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\section{Introduction}
The problem of estimating the support of a density has received a large amount of attention in the statistical literature since the
1980s
partly because of vast areas of applications in image analysis, signal processing and econometrics. The fundamental results in
this area were obtained in \cite{devroye1980detection, KorTsy1993, korostelev1993minimax, korostelev1994asymptotic, cuevas1997, CholaquidisFraiman}. Furthermore, see \cite{tsybakov1997nonparametric,walther1997, rigollet2009optimal,mason2009, Cholaquidis17} for a more general problem of estimating the level sets of a density.
In particular,
\cite{KorTsy1993} established the minimax optimal rates for estimating the support of a density having a H{\"o}lder-continuous boundary in the Hausdorff and
symmetric difference metrics and constructed
an estimator which attains the optimal rates. The case of convex support estimation was first studied in \cite{korostelev1994asymptotic,korostelev1993minimax}, where it was shown
that the convex hull \(\hat{C}\) of the sample points, which is the maximum likelihood estimator, is rate-optimal for estimating the support set \(C\) in the Hausdorff and
symmetric difference metrics.

The volume of a set is one of its most basic characteristics. Surprisingly, as it was shown in \cite{KorTsy1993, korostelev1993minimax},
the volume of a rate-optimal estimator of the set is not necessarily a rate-optimal
estimator of the volume. The first fully rate-optimal estimators of the volume of a convex support with smooth boundary and a support
with H{\"o}lder-continuous boundary  were
constructed by \cite{gayraud1997} based on three-fold sample splitting. A more efficient and flexible estimator
of the volume of a convex set with no assumption on the boundary was recently proposed in \cite{BaldinReiss16}.
In fact the proposed estimator is simple to compute and non-asymptotically efficient.
The problem of calculating the volume of a convex set has also attracted mathematicians working in
computer science and computational geometry, see \cite{dyer1991random,KanLovSim1997, lovasz2006simulated}.
The setting is slightly different: an experimenter uses a sampling algorithm that generates points over the space
that either fall inside or outside a set (this information is obtained via queries to the oracle).
We refer to \cite{vempala2010recent} for a recent survey of the existing fast randomised algorithms for calculating the volume of a convex set.
To the best of our knowledge, no efficient estimators
of the volume of a more general than convex class of sets have been rigorously studied.

\subsection{Main contribution and the structure}

In the present paper, we combine techniques from statistics and stochastic geometry to build a general framework for estimating
the volume of a set.
We focus on the Poisson point process (PPP) observation model with intensity $\lambda>0$ on a set $A$.
We thus observe
\begin{EQA}[c]
	X_1, ..., X_N \overset{i.i.d.}{\sim} U(A), \, \quad
	N \sim \text{Poiss}(\lambda\abs{A}),
\end{EQA}
where \((X_n), N\) are independent and \(|A|\) denotes the volume or the Lebesgue measure of the set \(A\). The set \(A\) is meant to belong to a class \({\mathbf A}\)
satisfying one simple assumption: \emph{the class is assumed to be intersection stable},
see Section~\ref{PPPTheory} for a concise definition.
The classes of sets covered by the assumption hence include:
\begin{itemize}
	\item convex sets;
	\item weakly-convex sets;
	\item star-shaped sets with a H{\"o}lder-continuous boundary;
	\item concentric sets;
	\item polytopes with fixed directions of outer unit normal vectors;
	\item compact sets.
\end{itemize}
In Section~\ref{PPPTheory},  we introduce
the so-called \emph{wrapping hull} \(\hat{A}\), which can informally be described as the minimal set from the class that
contains the data points \(X_1,...,X_N\).
It is then used in Section~\ref{OracleCaseSection} to construct the so-called oracle estimator for
the volume of a set belonging to one of the aforementioned classes when the intensity \(\lambda\) of the process  is known.
The oracle estimator is shown to be uniformly
of minimum variance among unbiased estimators (UMVU).
Section~\ref{DataDriveEstimation} is devoted to the estimation of the intensity and derives
a fully data-driven estimator of the volume,
\begin{EQA}[c]
	\label{FinEst0}
	\hat\vartheta = \frac{N + 1}{N_\circ + 1} |\hat{A}|\,,
\end{EQA}
where \(N_\circ\) is the number of data points lying in the interior of the wrapping hull. Figure~\ref{FigureIntro} illustrates an example in which a naive estimator \(|\hat{A}|\) significantly underestimates the true volume \(|A|\)  even in the case when the class of sets is known whereas the estimator \(	\hat\vartheta\) produces a rather striking performance, see Section~\ref{SimulationsSection} for a more detailed  numerical study\footnote{The simulations were implemented using the R package ``spatstat'' by \cite{spatstat}.}.
\begin{figure}[tp]
	\centering
	\includegraphics[width=\linewidth]{FigureIntro.png}
	\caption[]{On the left:  observations over an $r$-convex set \(A\), the annulus \(B(0.5,0.5)\setminus B(0.5,0.25)\) with the ball inside \( B(0.5,0.1)\). On the right: the $r$-convex hull \(\hat{A}\) (black) and the true set \(A\) (blue). The volume of the wrapping hull here is \( |\hat{A}| = 0.307\), the true volume is \(|A| = 0.620\) and our estimator yields \(	\hat\vartheta = 0.578\).}
	\label{FigureIntro}
\end{figure}
The mean squared risk of the estimator is shown to mimic the mean squared risk of the oracle estimator.
Although the main object of analysis is the PPP model, the key results
transfer to the so-called uniform model, cf. Section~\ref{UnifModelVolEst}, using ``Poissonisation''.
Section~\ref{ClassesOfBodies} further establishes the rates
of convergence of the oracle estimator and the estimator \(\hat\vartheta\) in \eqref{FinEst0} for the considered classes of sets satisfying the assumption.
Theorem~\ref{EfronsProp} states a generalized Efron's inequality  for the wrapping hull, cf. \cite{efron1965convex}, which reduces the analysis of the mean squared error of the
estimator \(\hat\vartheta\) to the distributional characteristics of the missing volume \(|A \setminus \hat{A}|\), a uniform lower bound on its expectation and a uniform
deviation inequality. Interestingly, a uniform lower bound on the expectation of the missing volume has not even been established
for the class of convex sets.
We therefore establish the rates of convergence only for a relatively simple class of polytopes with fixed directions of outer unit normal vectors in Section~\ref{PolFixeDirSec}. A more
general question is beyond the scope of the present paper and left to future research.
In volume estimation of weakly-convex sets in Section~\ref{RConnSect} there is a further peculiar question of adaptation
to a smoothing parameter.
We suggest an adaptation procedure inspired by Lepski's method, cf. \cite{lepskii1992asymptotically}, and study it numerically in Section~\ref{SimulationsSection}.
Our
numerical results in Section~\ref{SimulationsSection}, mainly devoted to volume estimation for the weakly-convex sets, in particular,
demonstrate that overestimating the smoothing parameter may have a significant cost for volume estimation.
Some of the technical lemmata are deferred to the Appendix.
Finally, we encounter and state a variety of new open questions in stochastic geometry, which we barely
begin to nibble at the edges. Interestingly enough, the framework was mentioned in a seminal paper \cite{kendall1974foundations} by David Kendall in the Statslab
at the University of Cambridge, but has never been fully explored.

\subsubsection{A simple one-dimensional example}
Let  \(X_1, ...,X_n\) be a sample of i.i.d. points drawn from the uniform distribution \(U(a,b)\), and
let \(X_{(1)},...,X_{(n)}\) denote the order statistics, so that
\(X_{(1)} < ... < X_{(n)}\).
It holds by symmetry that  the expected length of the interval \((X_{(1)}, X_{(n)})\) satisfies
\begin{EQA}[c]
	\label{EIAX}
	\mathbb{E} [|X_{(n)} - X_{(1)} |] =\frac{(n- 1)}{(n+1)} (b-a)\,.
\end{EQA}
An objective of statistical inference is to estimate the length of the interval, when  the location of the points \(a\) and \(b\) is assumed to be unknown.
A naive estimator,
\begin{EQA}[c]
	\hat{l}_{naive} := X_{(n)} - X_{(1)}\,,
\end{EQA} clearly
underestimates  the  length. A more attractive idea is to somehow dilate the interval \((X_{(1)}, X_{(n)})\)
and take the length of the dilated interval as an estimator.
There are at least two viable dilations: 1) add and subtract some fixed vectors
from the end points \(X_{(n)}\) and \(X_{(1)}\) (additive dilation) and 2) dilate the interval \((X_{(1)},X_{(n)})\)
from its centre \((X_{(n)} + X_{(1)}) / 2\) with some scaling factor (multiplicative dilation).
In the one-dimensional case, both dilations are equivalent.
It follows from \eqref{EIAX} that a reasonable additive dilation factor is \(2(X_{(n)} - X_{(1)})/(n-1)\) which yields  an estimator for the volume,
\begin{EQA}[c]
	\label{onedim}
	\hat{l}_{1} := \frac{(n+1)}{(n-1)}(X_{(n)} - X_{(1)})\,.
\end{EQA}

This estimator is not only \emph{unbiased},  \(\mathbb{E}[\hat{l}_{1}] = b - a\), but also, as we shall see in
Section~\ref{OracleCaseSection} and Section~\ref{DataDriveEstimation},
is minimax optimal.
We also refer to \cite{moore1984} for a comprehensive literature review of set estimation in the one-dimensional case.

\subsubsection{Estimation of the volume of a convex set in high dimensions}
The one-dimensional model is useful to grasp the main ideas of volume estimation, yet it is not
widely used in real applications. The two-dimensional model already covers several important applications in image analysis and signal processing.
Here, we observe the points \(X_1,...,X_n\) drawn uniformly over a set \(C \subset \mathbb{R}^2\) and
an objective is to recover the volume  \(V_C\) of the set and the set itself. Let us assume that
\(C\) belongs to the class of convex sets.
The wrapping hull is then simply the convex hull \(\hat{C}\) of the points \(\hat{C} = \text{conv}(X_1,...,X_n)\).
\begin{figure}[tp]
	\centering
	\begin{tikzpicture}
	\draw [dashed] (0,0) -- (-2,1) -- (-2.5,2) -- (-1.5,3.4) -- (0.5,3.8) -- (2.3, 3) -- (1.9, 1.5) -- (0,0);

	\draw [dashed,thick] (0.02734564, -0.6225691) -- (-2.58033532, 0.6812713) -- (-3.23225556,1.9851118) -- (-1.92841508,3.8104885) --
	(0.67926588, 4.3320247) -- (3.02617875,3.2889523) -- (2.50464256, 1.3331916) --  (0.02734564, -0.6225691);

	\draw [rotate around={15:(0,2)},pink] (0,2) ellipse (2.8 and 2.3);
	\draw [cyan, fill] (0,0) circle [radius=0.05];
	\draw [cyan, fill] (-2,1) circle [radius=0.05];
	\draw [cyan, fill] (-2.5,2) circle [radius=0.05];
	\draw [cyan, fill] (-1.5,3.4) circle [radius=0.05];
	\draw [cyan, fill] (0.5,3.8) circle [radius=0.05];
	\draw [cyan, fill] (2.3, 3) circle [radius=0.05];
	\draw [cyan, fill] (1.9, 1.5) circle [radius=0.05];
	\draw [black, fill] (-0.09, 2.049) circle [radius=0.05];
	\node at (-0.2, 1.8) {\(\hat x_0\)};

	\draw [purple, fill] (-1,2) circle [radius=0.05];
	\draw [purple, fill] (-0.5, 1) circle [radius=0.05];
	\draw [purple, fill] (-0.2, 0.4) circle [radius=0.05];
	\draw [purple, fill] (-1.4,1.7) circle [radius=0.05];
	\draw [purple, fill] (-0.4,3.2) circle [radius=0.05];
	\draw [purple, fill] (0.5,1) circle [radius=0.05];
	\draw [purple, fill] (0.7,3) circle [radius=0.05];
	\draw [purple, fill] (1,2) circle [radius=0.05];
	\draw [purple, fill] (1.6,2.5) circle [radius=0.05];

	\node at (2.1,4) {\(C\)};
	\node at (-1.4,3) {\(\hat{C}\)};
	\node at (3.2,3) {\(\tilde{C}\)};
	\node at (1.3,2.7) {\(X_1\)};
	\node at (0,0.6) {\(X_k\)};
	\node at (-1.5,1.5) {\(X_n\)};

	\end{tikzpicture}
	\caption{The points \(X_1,...,X_n\) drawn uniformly over a  set \(C\), the convex hull of the points \(\hat{C} = \text{conv}(X_1,...,X_n)\) and the dilated hull estimator
		\(\tilde{C}\).}
	\label{HB}
\end{figure}
Analogously to the one-dimensional case, it is natural to consider the volume \(|\hat{C}|\)
of the convex hull
as a baseline estimator for the volume \(V_C\) of the set \(C\). It is quite intuitive that this estimator performs quite poorly
because it always underestimates the true volume and it should therefore be dilated as in the one-dimensional case.
Section~\ref{OracleCaseSection} and Section~\ref{DataDriveEstimation} show that an optimal estimator has the following form
\begin{EQA}[c]
	\hat{V}_{opt} = \frac{n + 1}{n_\circ + 1} |\hat{C}|\,,
	\label{hvopt}
\end{EQA}
where \(n_\circ\) is the number of purple points in  Figure~\ref{HB} that lie in the interior of the convex hull \(\hat{C}\).
Note that \(\hat{V}_{opt}\) is the volume of the ``dilated'' hull \(\tilde{C}\), the set obtained by
dilating the convex hull with the same factor from the centre of gravity \(\hat x_0\) of the convex hull:
\begin{EQA}[c]
	\tilde{C} 	= \Big\{\hat x_0+\Bigl(\frac{n+1}{n_\circ+1}\Bigr)^{1/d}  (x-\hat x_0)\,\Big|\,x\in\hat{C}\Big\} \,,
\end{EQA}
which can in fact  be used to estimate the set \(C\) itself.
Similarly, the same estimators for the volume and the set itself can be used in higher dimensions.

The uniform model of a fixed number of  points drawn uniformly over a convex set \(C\) has been extensively studied in stochastic geometry.
The focus of study is rather on understanding the distributional characteristics of key functionals like the volume of \(\hat{C}\), the number of vertices of \(\hat{C}\) and
the distance between \(\hat{C}\) and \(C\). The main references here are \cite{barany1988convex, reitzner2003random, reitzner2005central, vu2005sharp, pardon2011}.
The Poisson point process (PPP) model studied in the present paper
is closely related to the uniform model. Using Poissonisation and de-Poissonisation techniques, this model exhibits asymptotic properties like the uniform model, see e.g. the references above and Section~\ref{UnifModelVolEst}.
However the geometric properties of the PPP model are much more fecund for conducting statistical inference, see
\cite{ReissSelk14,BaldinReiss16},
where the techniques from the Poisson point processes theory were successfully employed for estimation of linear functionals
in a one-sided regression model and estimation of the volume of a convex set.

\subsubsection{How fast can we estimate ${\pi}$?}
There are quite a few ways how one can calculate the number \(\pi\), see \cite{arndt2001pi}.
We here discuss one interesting way based on the Monte Carlo simulations of independent uniformly distributed random variables.
It is a toy illustrative application of volume estimation in sampling theory.
Let us draw the points \(X_1,...,X_N\) from the uniform distribution over the square \([0,1] \times [0,1]\)
and count the number of points \(n\) which fall inside the circle centred at the origin of radius \(1\).
Let \(\hat{\pi}:= n / N\) denote the ratio of the points inside the circle to the total number of points.
It approximately equals \(\pi/4\), because it is an unbiased estimator:
\begin{EQA}[c]
	\mathbb{E}[\hat{\pi}] = \frac{1}{N} \mathbb{E}[n] = \frac{1}{N} \mathbb{E}\big[\sum_{i = 1}^{N} {\bf 1}(X_i \in C)\big] = \frac{\pi}{4}\,,
\end{EQA}
and therefore its mean squared risk is governed  by the variance:
\begin{EQA}[c]
	\mathbb{E}\big[(\hat{\pi} - \pi)^2\big] = \text{Var}(\hat{\pi}) = \frac{1}{N^2} \text{Var}\big(\sum_{i = 1}^{N} {\bf 1}(X_i \in C)\big) = \frac{1}{N}\frac{\pi}{4} \big(1 - \frac{\pi}{4}\big)\,.
\end{EQA}
It turns out \(\hat{\pi}\) is even a maximum likelihood estimator. Surprisingly,  we are able to
estimate \(\pi\) with a much faster rate based on the data points in this experiment. Following \eqref{hvopt},
we define our properly scaled estimator as
\begin{EQA}[c]
	\hat{\pi}_{opt} = 4 \frac{n + 1}{n_\circ + 1} |\hat{C}|\,,
\end{EQA}
where \(n_\circ\) is the number of points lying inside the convex hull \(\hat{C}\) of the points lying inside the circle.
Theorem~\ref{UpBoundOracle} and Theorem~\ref{new_estimator_risk_wrap} to follow state that the
rate of convergence of the mean squared risk of the estimator \( \hat{\pi}_{opt} \) satisfies
\(\mathbb{E}\big[(\hat{\pi}_{opt} - \pi)^2\big] = \mathcal{O}(N^{-5/3})\), see Figure~\ref{pi} for a numerical comparison of the two estimators.
Note that both estimators can well be computed in polynomial time.
\begin{figure}[tp]
	\includegraphics[width=\linewidth]{pi2.png}
	\caption[]{On the left: a sample of \(n = 500\) points drawn uniformly over  the square \([0,1] \times [0,1]\).
		On the right: Monte Carlo root mean squared error (RMSE) estimates for the studied estimators for \(\pi\) based on 200 Monte Carlo
		simulations in each case.}
	\protect\label{pi}
\end{figure}

\subsection{Relationship to the work on volume estimation of a convex set}

Some of the theoretical results in the present paper are underpinned by the results for the convex case, cf.
Section~\ref{OracleCaseSection} and Theorem~\ref{new_estimator_risk_wrap} in
Section~\ref{DataDriveEstimation}, and the corresponding results  in \cite{BaldinReiss16}.
In fact,
a key observation igniting the development of the present framework is that estimation of the volume of convex sets
can in fact solely rely upon the property of convex sets being stable under taking intersections rather than convexity itself.
This striking observation appears to have a substantial value for volume estimation in a variety of scenarios far beyond convexity.
Volume estimation for some of the classes covered by the framework, in particular, the weakly-convex sets, has been long seen as notoriously
difficult with standard geometric arguments, see  the references in Section~\ref{ClassesOfBodies}.

Not violating the flow of the paper, we shall therefore omit some of the proofs
of the statements that are deduced from the proofs
of the corresponding statements for the convex case.
The proof of the result that the wrapping hull is a complete statistic in Theorem~\ref{SuffComplThm} is slightly simplified compared to the proof of the theorem that the convex hull is a complete statistic in  \cite{BaldinReiss16} and hinges  upon
a measure-theoretic result in stochastic geometry.
In contrast to the special case of convex sets, the present paper further argues
that the designed estimator \(\hat\vartheta \) is in fact \emph{adaptive} as its rate explicitly depends on
the rate of convergence of the missing volume \(|A \setminus \hat{A}|\).
This result rests upon Efron's inequality proved in Section~\ref{EfronWrapSect}.
Section~\ref{UnifModelVolEst} explicitly states that the same estimator is minimax optimal in the uniform model. Section~\ref{ClassesOfBodies} appears to convey the most noticeable value for applications as it provides efficient data-driven estimators and clearly outlines the steps of deriving explicit rates of convergence for specific classes of intersection stable sets.

\section{Poisson point process theory and the wrapping hull}
\label{PPPTheory}
In this section, we slightly digress on Poisson point processes, introduce the main notions and collect recently developed mathematical tools.
Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space, fix a convex compact set \({\mathbf E} \) in \(\mathbb{R}^d\)
and equip it with the Borel \(\sigma\)-algebra \(\mathcal{E}\) with respect to the Euclidean metric \(\rho\). Without loss of generality one may assume that \({\mathbf E} = [0,1]^d\).
By a \emph{point process} \(\mathcal{N}\) on \({\mathbf E}\) we mean an integer-valued random measure, or a kernel,
from the probability space \((\Omega, \mathcal{F}, \mathbb{P})\) into \(({\mathbf E}, \mathcal{E})\). Thus, \(\mathcal{N}\)
is a mapping from \(\Omega \times \mathcal{E}\)  into \(\{0,1,...\}\) such that \(\mathcal{N}(\omega, \cdot)\) is
an integer-valued measure for fixed \(\omega \in \Omega\) and \(\mathcal{N}(\cdot, B)\) is  an integer-valued random variable
for fixed \(B \in \mathcal{E}\). For convenience, we write \(\mathcal{N}(B) = \mathcal{N}(\cdot, B)\).

Let \({\mathbf K}\) be the set of all compact subsets of \({\mathbf E}\) equipped with its
Borel \(\sigma\)-algebra \(\mathfrak{B}_{{\mathbf K}}\)
with respect to the Hausdorff-metric $\rho_H$ defined for two non-empty compact sets \(A\) and \(B\) by
\begin{EQA}[c]
	\rho_H(A,B) = \max \big( \sup_{x \in B} \rho(x, A), \sup_{x \in A} \rho(x, B)\big)\,.
\end{EQA}
It is known, see Theorem C.5 in \cite{molchanov2006}, that the  Borel \(\sigma\)-algebra \(\mathfrak{B}_{{\mathbf K}}\) coincides with
the \(\sigma\)-algebra \(\sigma([B]_{{\mathbf K}}, B \in {\mathbf K})\) with \([B]_{{\mathbf K}} = \{A \in {\mathbf K} : A \subseteq B \}\).  Moreover, the space \(({\mathbf K}, \rho_H) \) is Polish.

\vspace{10pt}
Let \({\mathbf A} \subset {\mathbf K}\) be a family of compact subsets of \({\mathbf E}\) fulfilling  the following assumption
\begin{assumption}
	\label{Assumption}
	\({\mathbf A}\) is closed under taking arbitrary intersections and \(\varnothing, {\mathbf E} \in {\mathbf A}\).
\end{assumption}
\vspace{10pt}

Then the metric subspace \(({\mathbf A}, \rho_H)\) has
the induced Borel \(\sigma\)-algebra \(\mathfrak{B}_{\mathbf A} = {\mathbf A} \cap \mathfrak{B}_{\mathbf K} = \{{\mathbf A} \cap K : K \in \mathfrak{B}_{\mathbf K} \}\), which thus
coincides with the \(\sigma\)-algebra
\(\mathcal{A} = \sigma([B], B \in {\mathbf A})\) where \([B] = \{A \in {\mathbf A}: A \subseteq B \}\).
It turns out there is a fascinating connection between the families of sets satisfying Assumption~\ref{Assumption} and the \emph{trapping systems} introduced in the groundbreaking  work of \cite{kendall1974foundations} on the theory of random sets, see also Section 7.2 in \cite{molchanov2006}.

We call a point process \(\mathcal{N} : \Omega \to M({\mathbf E})\) a Poisson point process (PPP) of intensity \(\lambda > 0 \) on \(A \in {\mathbf A}\), if
\begin{itemize}
	\item for any \(B \in  \mathcal{E}\), we have \(\mathcal{N}(B) \sim \text{Poiss}\bigl(\lambda |A\cap B|\bigr)\), where $\abs{A\cap B}$ denotes the Lebesgue measure of $A\cap B$;
	\item for all mutually disjoint sets \(B_1,..., B_n \in    \mathcal{E}\), the
	random variables \newline \(\mathcal{N}(B_1),...,\mathcal{N}(B_n) \) are independent.
\end{itemize}
For statistical inference, we assume the Poisson point process to be defined on a set of non zero  Lebesgue measure, i.e. \(|A| >\delta\) for
\(\delta >0\).

Interestingly, one can view \( (\mathcal{N}(K),K \in {\mathbf K})\) as a set-indexed stochastic process. It has no direct order
and its natural filtration is defined by
\begin{EQA}[c]
	\mathcal{F}_K \stackrel{\operatorname{def}}{=} \sigma(\{\mathcal{N}(U); U \subseteq K,  U \in {\mathbf K} \})
\end{EQA}
for any \( K \in {\mathbf K}\). The properties of  the filtration \((\mathcal{F}_K, K \in {\mathbf K})\) are well studied, cf
\cite{zuyev1999stopping}. By construction, the restriction \(\mathcal{N}_K = \mathcal{N}(\cdotp \cap K)\) of the point process \(\mathcal{N}\) onto
\(K \in \mathbf{K}\) is \(\mathcal{F}_K\)-measurable (in fact, \(\mathcal{F}_K = \sigma(\{\mathcal{N}_K(U); U \in \mathbf{K}\})\)). Moreover, it can be easily seen that \(\mathcal{N}_K\) is a Poisson point process in \(\mathbf{M}\), cf. the
Restriction Theorem in \cite{kingman1992poisson}.
Furthermore, we say that a set-indexed, \((\mathcal{F}_K)\)-adapted integrable process
\((X_K, K \in {\mathbf K})\) is a martingale if  \(\mathbb{E}[X_B| \mathcal{F}_A] = X_A\) holds for any \(A, B \in {\mathbf K}\) with \(A \subseteq B\).
By the independence of increments, the process
\begin{EQA}[c]
	M_K \stackrel{\operatorname{def}}{=} \mathcal{N}(K) - \lambda|K|,\quad K\in{\mathbf K}\,,
\end{EQA}
is clearly a martingale with respect to its natural filtration \((\mathcal{F}_K,K\in{\mathbf K})\).

A random compact set \(\mathcal{ K}\) is a measurable mapping \(\mathcal{ K}:  (\mathbf{M},\mathcal{M}) \to ({\mathbf K},\mathfrak{B}_{\mathbf K})\).
A random compact set \(\mathcal{ K}\) is called an \(\mathcal{F}_K\)-stopping set if \(\{\mathcal{ K} \subseteq K\} \in \mathcal{F}_K\)
for all \(K \in {\mathbf K}\). The sigma-algebra of \(\mathcal{ K}\)-history is defined as
$\mathcal{F}_{\mathcal{ K}} = \{A \in \mathfrak{F}: A \cap \{ \mathcal{ K} \subseteq K \} \in \mathcal{F}_K\,\,\, \forall K \in {\mathbf K} \}$, where
\(\mathfrak{F} =   \sigma(\mathcal{F}_K; K \in {\mathbf K})\).
We introduce the \emph{wrapping hull} of the PPP points on a set \(A \in {\mathbf A}\), which is served as a set-estimator of \(A\).

\vspace{10pt}
\begin{definition}
	\label{WrapDef}
	The \({\mathbf A}\)-\emph{wrapping hull} (or simply the wrapping hull) of the PPP points is a mapping \(\hat{A}: M \to {\mathbf A}\) defined as
	\begin{EQA}[c]
		\hat{A} \stackrel{\operatorname{def}}{=} \text{wrap}_\Ab\{X_1,...,X_N\} \stackrel{\operatorname{def}}{=} \bigcap \{A \in {\mathbf A}:  X_i \in A, \forall i= 1,...,N \}\,.
	\end{EQA}
\end{definition}
\vspace{10pt}

For a set $A\subset {\mathbf E}$ let $A^c$ denote its complement.

\begin{lemma}
	\label{LemmaStoppingSet}
	The set $\hat\mathcal{ K}\stackrel{\operatorname{def}}{=}\overline{\hat{A}^c}$, the closure of the complement of the wrapping hull, is an $(\mathcal{F}_K)$-stopping set.
\end{lemma}
The proof of Lemma~\ref{LemmaStoppingSet} essentially repeats the steps of the proof of an analogous statement for the convex hull of the PPP points given in Lemma 2.2 in \cite{BaldinReiss16}. A striking observation is the following corollary, which rests upon
the optional sampling theorem for set-indexed martingales, cf. \cite{zuyev1999stopping}.
\begin{corollary}
	\label{EfronsIdenPoisson}
	The number of points \(N_\partial\) lying on the wrapping hull \(\hat{A}\) and
	the missing volume \(|A \setminus \hat{A}|\) satisfy the relation
	\begin{EQA}[c]
		\label{efronsIdent2}
		\mathbb{E}[N_\partial] = \lambda\, \mathbb{E}[ |A\setminus \hat{A}|]\,.
	\end{EQA}
\end{corollary}

We shall define the \emph{likelihood function} for the PPP model. Note that we can evaluate the probability
\[
\mathbb{P}_A\bigl(\hat{A} \in  B\bigr)
= \sum_{k = 0}^{\infty} \frac{{\rm e}^{-\lambda \abs{A}} \lambda^k}{k!}
\int_{A^k} {\bf 1}(\text{wrap}_\Ab\{x_1, ..., x_k\} \in B ) {\rm d} (x_1,..., x_k)
\]
for \(B \in \mathfrak{B}_{{\mathbf A}}\). Usually, we only write the subscript $A$ or sometimes $(A,\lambda)$ when different probability distributions are considered simultaneously.
The likelihood function \(L(A, \mathcal{N}) = \frac{{\rm d} \mathbb{P}_{A,\lambda}}{{\rm d} \mathbb{P}_{\mathbf{E},\lambda_0}}\) for \(A \in {\mathbf A}\) and $\lambda,\lambda_0>0$
is then given by
\begin{EQA}[c]
	L(A, \mathcal{N}) = \frac{{\rm d} \mathbb{P}_{A,\lambda}}{{\rm d} \mathbb{P}_{\mathbf{E},\lambda_0}}(\mathcal{N})

	= {\rm e}^{\lambda_0|{\mathbf E}|-\lambda|A|}(\lambda/\lambda_0)^N {\bf 1}(\hat{A} \subseteq A)\,,
	\label{fPPeb}
\end{EQA}
cf. Thm. 1.3 in \cite{Kutoyants1998}. For the last line, we have used that a point set is in $A$ if and only if its wrapping hull
\(\hat{A} = \text{wrap}_\Ab\{X_1,...,X_N\}\) is contained in $A$.

The following theorem is an essential ingredient for deriving our estimators.
The statement
of the theorem is quite intuitive and already used in \cite{privault2012invariance}. Its proof is similar to the proof of Theorem 3.1 in \cite{BaldinReiss16}.
\begin{theorem}
	\label{LemMeasCond}
	Set $\hat\mathcal{ K}\stackrel{\operatorname{def}}{=}\overline{\hat{A}^c}$.
	The number \(N_{\partial}\) of points on the boundary of the wrapping hull \(\hat{A}\) is measurable with respect to the sigma-algebra of
	\(\hat\mathcal{ K}\)-history \(\mathcal{F}_{\hat\mathcal{ K}}\). The number of points in the interior of the wrapping hull \(N_\circ\) is, conditionally on \(  \mathcal{F}_{\hat\mathcal{ K}}\), Poisson-distributed:
	\begin{equation}
	\label{NccCs}
	N_\circ \, \big| \, \mathcal{F}_{\hat\mathcal{ K}} \sim \text{Poiss}(\lambda_\circ)\text{ with }\lambda_\circ \stackrel{\operatorname{def}}{=} \lambda | \hat{A} |.
	\end{equation}
	In addition, we have \(\mathcal{F}_{\hat\mathcal{ K}} = \sigma(\hat{A})\), where
	the latter is the sigma-algebra \( \sigma(\{\hat{A} \subseteq B, B \in \mathcal{A}\})\) completed with the null sets in  \(\mathfrak{F}\).
\end{theorem}

We shall further use the following short notation:  \(N =\mathcal{N}({\mathbf E})\) denotes the total number of points, \(N_\circ = \mathcal{N}({\hat{A}}^\circ)\) the number of points in the interior
of the wrapping hull \(\hat{A}\) and \( N_{\partial} = \mathcal{N}(\partial\hat{A})=\mathcal{N}(\partial\hat\mathcal{ K}) \) the number of points on the boundary of the wrapping hull.
For asymptotic bounds we write \(f(x) = O(g(x))\) or \(f(x) \lesssim g(x)\) if \(f(x)\) is bounded by a constant multiple of \(g(x)\)  and
\(f(x) \thicksim g(x)\) if \(f(x) \lesssim g(x)\)  as well as \(g(x) \lesssim f(x)\).

\section{Oracle case: known intensity $\lambda$}\label{OracleCaseSection}
In the case when \(\lambda\) is known one can just estimate the volume \(|A|\) by \(N/\lambda\), which is an unbiased estimator,
whose mean squared risk is given by
\begin{EQA}[c]
	\label{RiskNaiveEst}
	\mathbb{E}[(N / \lambda - |A|)^2] = \text{Var}(N/\lambda) = |A| / \lambda \,,
\end{EQA}
thus implying \(\mathcal{O} (\lambda^{-1/2})\)-rate of convergence. This rate can be improved.
As we shall see in Theorem~\ref{SuffComplThm}, the wrapping hull is a complete and sufficient statistic thus allowing
to construct the unique best unbiased estimator of the volume in virtue of the Lehmann-Scheff\'e theorem.
In view of Theorem~\ref{LemMeasCond} we thus derive our oracle estimator
\begin{EQA}[c]
	\hv_{\text{oracle}}=
	\mathbb{E} \Big[\frac{N}{\lambda} | \mathcal{F}_{\hat\mathcal{ K}} \Big]  =  \mathbb{E} \Big[\frac{N_\circ + N_\partial}{\lambda} | \mathcal{F}_{\hat\mathcal{ K}} \Big] = \frac{N_\partial}{\lambda} + |\hat{A}|
	\label{OracleEst}
\end{EQA}
The following result is fundamental in characterising the rate of convergence of the risk of the oracle estimator.
\begin{theorem}
	\label{UpBoundOracle} For known intensity \(\lambda > 0\), the oracle estimator \(\hv_{\text{oracle}}\) is unbiased and of minimal variance among all unbiased estimators (UMVU)
	in the PPP model with parameter class \({\mathbf A}\). It satisfies
	\begin{EQA}[c]
		\label{ORACLEEst}
		\mathbb{E} \big[(\hv_{\text{oracle}}- |A|)^2\big] = \text{Var}(\hv_{\text{oracle}}) = \frac{\mathbb{E}[|A\setminus \hat{A}|]}{\lambda}\,.
	\end{EQA}
\end{theorem}
\begin{remark} The theorem  asserts that the rate of convergence of \(\hv_{\text{oracle}}\) is in fact faster than \(\lambda^{-1/2}\) for all classes of sets \({\mathbf A}\) satisfying
	Assumption~\eqref{Assumption}.
\end{remark}

\begin{proof}
	By the tower property of conditional expectation, the estimator \(\hv_{\text{oracle}}\) is unbiased, \(\mathbb{E}[\hv_{\text{oracle}}] = |A|\). Using law of total variance, we derive
	\begin{EQA}
		\text{Var}(\hv_{\text{oracle}}) = \text{Var}\Big(\mathbb{E} \Big[\frac{N}{\lambda} | \mathcal{F}_{\hat\mathcal{ K}} \Big]\Big)& =& \text{Var}(\frac{N}{\lambda}) - \mathbb{E} \Big[\text{Var}\Big(\frac{N}{\lambda}| \mathcal{F}_{\hat\mathcal{ K}} \Big)\Big] \\
		&=& \frac{|A|}{\lambda} - \mathbb{E} \Big[\text{Var}\Big(\frac{N_\circ + N_\partial}{\lambda}|\mathcal{F}_{\hat\mathcal{ K}} \Big)\Big] = \frac{\mathbb{E}[|A\setminus \hat{A}|]}{\lambda}\,.
	\end{EQA}
	Theorem~\ref{SuffComplThm} below affirms that the wrapping hull \(\hat{A}\) is a
	complete and sufficient statistic such that by the Lehmann-Scheff\'e theorem, the
	estimator \(\hv_{\text{oracle}}\) has the UMVU property.
\end{proof}

\begin{theorem}
	\label{SuffComplThm} For known intensity \(\lambda > 0\), the wrapping hull  is a complete and sufficient statistic.
\end{theorem}
The proof of Theorem~\ref{SuffComplThm} is deferred to the Appendix.
As a result of Theorem~\ref{UpBoundOracle}, the performance of the estimator \(\hv_{\text{oracle}}\) of the volume is reduced to the analysis of the performance of the wrapping hull
estimator of the set itself, which clearly depends on the geometric properties of classes of sets satisfying
Assumption~\ref{Assumption}.

\begin{comment}
\subsection{Minimax upper bound in terms of the metric entropy}
This section shows that the minimax upper bound on the risk  of the oracle estimator
\(\hv_{\text{oracle}}\) in Theorem~\ref{UpBoundOracle}
hinges solely upon the geometric structure, specifically the metric entropy, of an underlying class of sets.
Recall that an \emph{\(\epsilon\)-net} of a class \({\mathbf A}\) with respect to a metric \(\rho\) is a collection \(\{A^1,...,A^N \} \in {\mathbf A}\)
such that for each \(A \in {\mathbf A}\), there exists \(i \in \{1,...,N\} \) such that \(\rho(A,A^i) \le \epsilon\). The \emph{\(\epsilon\)-covering
number} \(N({\mathbf A}, \rho, \epsilon)\) is the cardinality of the smallest \(\epsilon\)-net. The \emph{\(\epsilon\)-entropy} of the class \({\mathbf A}\) is
defined by \(H({\mathbf A}, \rho, \epsilon) = \log_2 N({\mathbf A}, \rho, \epsilon)\).
For some intersection stable classes of sets, it appears  easier to upper bound the metric entropy and hence derive a bound on the rate of convergence of the oracle estimator rather
than tackling the missing volume directly. The proof of the following theorem is deferred to the Appendix.
\begin{theorem}
\label{UpperBoundEntropy}
Let \( \epsilon_\lambda\) satisfy \(\epsilon_\lambda = H_{\epsilon_\lambda} / \lambda := H({\mathbf A}, \rho_1, \epsilon) /\lambda\).
Then, in the PPP model, the mean squared risk of the oracle estimator satisfies
\begin{EQA}[c]
\sup_{A \in {\mathbf A}, |A| >0} \mathbb{E} \big[(\hv_{\text{oracle}}- |A|)^2\big] \lesssim \epsilon_\lambda/ \lambda \, .
\end{EQA}
\end{theorem}

\subsection{A generic scheme of deriving the minimax lower bounds}

\begin{remark}
Interestingly, Theorem~\ref{UpperBoundEntropy} and Theorem yield that the minimax
rate of volume estimation is always in the interval \([c\lambda^{-2}, \lambda^{-1}]\) for some universal constant \(c>0\).
\end{remark}

\end{comment}

The minimax lower bounds on the rate of convergence of the risk of estimating the volume of a  set \(A \in {\mathbf A}\)  are often easier to establish for concrete classes of sets using the so-called hypercube argument, cf.  \cite{gayraud1997}. Interestingly, the following general bound on the minimax optimal rate holds.
\begin{theorem}
	\label{MinimaxLowerBoundThm}
	The minimax optimal rate of estimating the volume of a set \(A \in {\mathbf A}\)  satisfies
	\begin{EQA}[c]
		\label{eq:ddvvvbe}
		\lambda^{-2} \lesssim	\inf_{ \hat{\vartheta}_\lambda} \sup_{A \in {\mathbf A}}	\mathbb{E} \big[(\hat{\vartheta}_\lambda - |A|)^2\big] \lesssim \lambda^{-1}\,,
	\end{EQA}
	where the infimum  extends over all estimators \(\hat{\vartheta}_\lambda \) in the Poisson point process model with intensity \(\lambda\).
\end{theorem}
\begin{remark} The rate \(	\mathcal{O}(\lambda^{-1})\) is minimax for estimating the volume in some parametric classes of sets, in particular, the class of concentric sets, whereas the rate \(\mathcal{O}( \lambda^{-1/2})\) is established for estimating the volume in the class of compact sets, see Section~\ref{ClassesOfBodies}.
\end{remark}
\begin{proof}
	The upper bound in \eqref{eq:ddvvvbe} follows directly from Theorem~\ref{UpBoundOracle}.
	The lower bound is obtained by reducing the minimax risk to the Bayes risk and then lower-bounding the Bayes risk at its minimum. These steps are fairly standard,
	cf. \cite{korostelev1993minimax}, and we hence omit them here.
\end{proof}

\section{Data-driven estimator of the volume}
\label{DataDriveEstimation}
The main ingredient to deriving the estimator of \(\lambda\) is the fact that
the closure of the complement of the \({\mathbf A}\)-wrapping hull \(\hat\mathcal{ K}\stackrel{\operatorname{def}}{=}\overline{\hat{A}^c}\)
is in fact an $(\mathcal{F}_K)$-stopping set according to Lemma~\ref{LemmaStoppingSet}.
Moreover in analogy with a time-indexed Poisson process,
our problem boils down to the estimation of the intensity of a time-indexed Poisson process
starting from an unknown origin. To see this, recall that  according to Theorem~\ref{LemMeasCond},
the number of points \(N_\circ\) lying inside the wrapping hull \(\hat{A}\) is Poisson-distributed with intensity \(\lambda_\circ \stackrel{\operatorname{def}}{=} \lambda | \hat{A} |\)
provided that \(|\hat{A}| > 0\):
\begin{EQA}[c]
	N_\circ \, \big| \, \mathcal{F}_{\hat\mathcal{ K}} \sim \text{Poiss}(\lambda_\circ).

\end{EQA}
We aim to find an estimator for \(\lambda_\circ^{-1}\). On the event \(|\hat{A}| > 0\), we follow the idea developed in \cite{BaldinReiss16}.
That is to say, we use that the first jump time \(\tau\) of a time-indexed Poisson process \((Y_t, t > 0)\) with intensity \(\nu > 0\)
is \(\text{Exp}(\nu)\)-distributed and hence \(\mathbb{E}[\tau] = \nu^{-1}\). Using the memoryless property of the
exponential distribution, we then have
\begin{EQA}[c]
	\mathbb{E}[\tau | Y_1 = m] = \frac{1}{m+1}{\bf 1}(m \ge 1)+ (1 + \nu^{-1}){\bf 1}(m = 0)\,.
\end{EQA}
Therefore, we conclude that
\[ \hat\mu(N_\circ, \lambda_\circ) \stackrel{\operatorname{def}}{=} \begin{cases} (N_\circ+1)^{-1},& \text{ for }N_\circ\ge 1,\\ 1+\lambda_\circ^{-1},&\text{ for } N_\circ=0
\end{cases}
\]
satisfies \(\mathbb{E}[\hat\mu(N_\circ, \lambda_\circ) | \mathcal{F}_{\hat\mathcal{ K}}] = \lambda_\circ^{-1}\) provided that \(|\hat{A}| > 0\). Omitting the term
depending on \(\lambda_\circ\) in the unlikely event \(N_\circ = 0\), we derive our final estimator:
\begin{EQA}[c]
	\label{FinEst}
	\hat\vartheta = |\hat{A}| + \frac{N_\partial}{N_\circ + 1} |\hat{A}| = \frac{N + 1}{N_\circ + 1} |\hat{A}|\,.
\end{EQA}
\begin{remark}
	As it follows from Definition~\ref{WrapDef}, a wrapping hull \(\hat{A}\) may consist of disjoint sets, in which case
	the number of points \(N_{\circ,k}\) lying inside a piece \(\hat{A}_k\) satisfies
	\(N_{\circ,k} \, \big| \, \mathcal{F}_{\hat\mathcal{ K}} \sim \text{Poiss}(\lambda |\hat{A}_k|)\) due to the homogeneity of the Poisson point process.
	This fact can further be used to estimate \(\lambda^{-1}\) locally. However, in the homogeneous case,
	we prefer to use the total number of points to estimate the intensity.
\end{remark}

Note that a more explicit bound can be derived using the Cauchy-Schwarz inequality given  a bound on the
expected number of points \(N_{\partial}\) lying on the wrapping hull \(\hat{A}\) and  a bound on the moments of the missing volume \( | A \setminus \hat{A}|\).
This clearly depends on a considered class of sets satisfying Assumption~\ref{Assumption}. The following very general oracle inequality holds
for the mean squared error of the estimator \( \hat\vartheta\). Its proof
 can be adapted from the proofs of Thm.~4.4 and Thm.~4.5  in \cite{BaldinReiss16} and we hence omit it here.

\begin{theorem}
	\label{new_estimator_risk_wrap}
	The following oracle inequality for the risk of the estimator \( \hat\vartheta\) holds
	for all $A \in {\mathbf A}$ whenever $\lambda\abs{A}\ge 1$:
	\begin{EQA}[c]
		\mathbb{E}[(\hat\vartheta - |A|)^2]^{1/2} \le  (1+c \alpha(\lambda, A))\text{Var}(\hat\vartheta_{oracle})^{1/2} +r(\lambda,A)\,,
	\end{EQA}
	where
	\begin{align*}
	\alpha(\lambda, A) &:= \frac{1}{\abs{A}} \Bigl(\frac{ \text{Var}(\abs{A\setminus\hat{A}} )}{ \mathbb{E}[ \abs{A\setminus\hat{A}}]}
	+   \mathbb{E}[ \abs{A\setminus\hat{A}}] \Bigr)\,,\\
	r(\lambda,A) &:=c_1 \lambda^{-1} \mathbb{E}[N_{\partial}^4]^{1/4} \mathbb{P}(|A\setminus \hat{A}| \ge |A|/2)^{1/4}\,,
	\end{align*}
	with some numeric constants \(c,c_1 > 0\). In particular, \(\alpha(\lambda, A)\) is bounded by some universal constant.
\end{theorem}

\subsection{Volume estimation in the uniform model}
\label{UnifModelVolEst}
In the PPP model, the data we observe are uniformly distributed points over a set in some given class and the number of points is a realisation of a Poisson random variable. The uniform model,
\begin{EQA}[c]
	X_1, ..., X_n \overset{i.i.d.}{\sim} U(A), \, \quad A \in {\mathbf A}\,,
\end{EQA}
is closely related to the PPP model and assumes that the number of points \(n\) is fixed. In stochastic geometry, the objects studied in the PPP model typically exhibit a similar asymptotic behaviour in the uniform model and vice versa, see e.g. \cite{pardon2011} and references therein for a study of the functionals of the convex hull. This section
examines which results of the present paper derived in the PPP model remain true in the uniform model.

It is relatively straightforward to show that the wrapping hull remains a sufficient and complete statistic in the uniform model with slightly adjusted arguments of the proof of Theorem~\ref{SuffComplThm}. It is unknown however
whether there exists an UMVU estimator in the uniform model. Nevertheless, an estimator
\begin{EQA}[c]
	\label{FinEstUniform}
	\hat\vartheta_{\text{unif},n} := \frac{n + 1}{n_\circ + 1} |\hat{A}|\,,
\end{EQA}
where \(n_\circ \) is the number of points lying inside the wrapping hull, inherits the
same rate of convergence as the final estimator \(\hat\vartheta\) in \eqref{FinEst} in the PPP model due to the following result
\begin{proposition}[Poissonisation]
	Let \(n = \lfloor \lambda |A| \rfloor > 0\) with \(A\) being any set in a class \({\mathbf A}\).
	Then letting \(\lambda \to \infty\) the following asymptotic equivalence result holds true
	\begin{EQA}[c]
		\label{PoissDePoissMinMax}
		\mathbb{E} \big[(\hat\vartheta_{\text{unif},n} - |A|)^2\big] \thicksim
		\mathbb{E} \big[(\hat\vartheta - |A|)^2\big] \,, \quad \forall A \in {\mathbf A}\,.
	\end{EQA}
	Furthermore, the minimax lower bounds satisfy
	\begin{EQA}[c]
		\label{PoissDePoissMinMax2}
		\inf_{ \hat{\vartheta}_n} \sup_{A \in {\mathbf A}}	\mathbb{E} \big[( \hat{\vartheta}_n - |A|)^2\big] \thicksim
		\inf_{ \hat{\vartheta}_\lambda} \sup_{A \in {\mathbf A}}	\mathbb{E} \big[(\hat{\vartheta}_\lambda - |A|)^2\big] \,,
	\end{EQA}
	where the infimum on the left-hand side extends over all estimators in the uniform model, whereas the infinum on the right-hand side extends over all estimators in the Poisson point process model.
\end{proposition}

\begin{proof}
	We only prove \eqref{PoissDePoissMinMax2} here. \eqref{PoissDePoissMinMax} can then be proved exploiting similar arguments. Let us first show the inequality \( "\gtrsim" \). Assume it does not hold and that there exists an estimator \(\hat{\vartheta}_n^\prime \) in the uniform model with the rate of convergence faster than the minimax optimal rate in the PPP model. Then for the estimator
	\(\hat{\vartheta}_{N}^\prime \) we have for any \(A \in {\mathbf A}\)
	\begin{EQA}
		\mathbb{E} \big[(\hat{\vartheta}_{N}^\prime  - |A|)^2\big] & =& \sum_{k = 1}^{\infty } \mathbb{E} \big((\hat{\vartheta}_{N}^\prime  - |A|)^2 \, \big| \, N = k\big)\, \mathbb{P}(N = k) \\ \\
		&\le& \sum_{k =  \lfloor \lambda |A|/2 \rfloor }^{ \lfloor 2 \lambda |A| \rfloor  } \mathbb{E} \big((\hat{\vartheta}_{N}^\prime  - |A|)^2 \, \big| \, N = k\big)\, \mathbb{P}(N = k)  + c_2 \exp(-c_3 n )\\
		&\le& c_1 \mathbb{E} \big[( \hat{\vartheta}_n^\prime - |A|)^2\big]  + c_2 \exp(-c_3 n)\,,
	\end{EQA}
	for some constants \(c_1,c_2,c_3 > 0\) using Bennett's inequality, a contradiction in view of Theorem~\ref{MinimaxLowerBoundThm}. The other direction follows using the same technique.
\end{proof}

\subsection{Efron's inequality  for the wrapping hull}
\label{EfronWrapSect}
In this section, we show that  the rate of convergence of the risk for the estimator \(\hat\vartheta\) in Theorem~\ref{new_estimator_risk_wrap}
hinges in fact upon only a deviation of the missing volume  \(| A \setminus \hat{A}|\).
More than 50 years ago Efron showed  in \cite{efron1965convex} that the moments of the number of the points \(N_{{\partial},k}\) lying on the
boundary of a convex hull \(\hat{C}_{k}\) in the uniform model \(X_1,...,X_k \overset{i.i.d.}{\sim} U(C)\), with \(C \subset \mathbb{R}^d\)
being a convex set, satisfies the identity
\begin{EQA}[c]
	\mathbb{E}[N_{{\partial},k}^q] = \sum_{r = 1}^{q} n(k, q, r) \mathbb{E}[|\hat{C}_{k-r}|^r]\,,
\end{EQA}
where \(n(k, q, r)\) is the number of \(q\)-tuples from \(1,...,k\) having exactly \(r\) different values,
\(n(k, q, r) = {k \choose r} \sum_{m = 1}^{r} (-1)^{r - m}  {r \choose m} m^q\). This yields a striking dimension-free
asymptotic equivalence result,
\begin{EQA}[c]
	\mathbb{E}[N_{{\partial},k}^q] \thicksim k^q \mathbb{E}[|C \setminus \hat{C}_{k}|^q]\,.
	\label{EfronsIneqRev}
\end{EQA}

We here extend a one-sided version of this results to the wrapping hull.
\begin{proposition}[Efron's inequality  for the wrapping hull]
	\label{EfronsProp}
	Let \({\mathbf A}\) be any class satisfying Assumption~\ref{Assumption} and \(\hat{A}\) be the corresponding wrapping hull of the PPP points
	of intensity \(\lambda > 0\)
	over a set \(A \in {\mathbf A}\).
	Then the following asymptotic inequality holds
	\begin{EQA}[c]
		\mathbb{E}[N_{\partial}^q] \lesssim \lambda^q \mathbb{E}[|A \setminus \hat{A}|^q]\,,
	\end{EQA}
	provided that the probability of observing \(q\) points lying on the boundary of the wrapping hull \(\hat{A}\) is non-zero.
\end{proposition}
\begin{remark}
	It follows by Jensen's inequality and Corollary~\ref{EfronsIdenPoisson}, that
	\(\mathbb{E}[N_{\partial}^q]  \ge \mathbb{E}[N_{\partial}]^q = \lambda^q \mathbb{E}[|A \setminus \hat{A}|]^q\). For some examples,
	like the class of convex sets, this in fact implies \(\mathbb{E}[N_{\partial}^q] \thicksim \lambda^q \mathbb{E}[|A \setminus \hat{A}|^q]\).
\end{remark}
\begin{remark}
	Identities that relate the functionals of the convex hull of the points distributed uniformly over a convex set are thoroughly studied in stochastic geometry, see \cite{pardon2011, buchta2013,reitzner2015}.

\end{remark}

\begin{proof}
	Let us first consider the uniform model and then transfer the result to the PPP model using Poissonisation.
	We follow Efron's idea, see also \cite{reitzner2015, brunel:tel-01066977}, that
	\begin{EQA}
		\mathbb{E}[|A \setminus \hat{A}_k|^q] &=& |A|^q \, \mathbb{P}(X_{k + 1} \notin \hat{A}_{k},..., X_{k + q} \notin \hat{A}_{k} )  \\
		& \ge &  |A|^q\, \mathbb{P}(X_{k + 1} \in \partial \hat{A}_{k+q},..., X_{k + q} \in \partial \hat{A}_{k+q}) \\
		& = &  \frac{|A|^q}{{k + q \choose q} }\,  \mathbb{E} \sum_{}^{} {\bf 1}(X_{i_1} \in \partial \hat{A}_{k+q},..., X_{i_q} \in \partial \hat{A}_{k+q}) \\
		&=& \frac{|A|^q}{{k + q \choose q} }\,  \mathbb{E} {N_{{\partial}, k+q} \choose q}\,,
	\end{EQA}
	the sum being taken over all tuples \((i_1, ..., i_q)\) from the integers \(1,..., k+q\). Rearranging the terms, this entails
	\( \mathbb{E} [N_{\partial, k}^q] \lesssim k^q \mathbb{E}[|A \setminus \hat{A}_k|^q] \).

	Using Poissonisation, we further derive for the PPP model,
	\begin{EQA}[c]
		\mathbb{E}[|A \setminus \hat{A}|^q] = \sum_{k = 1}^{\infty } \mathbb{E}(|A \setminus \hat{A}_N|^q \, \big| \, N = k)\, \mathbb{P}(N = k) \\
		\gtrsim \sum_{k = 1}^{\infty } (2 \lambda|A|)^{-q} \mathbb{E}\big[ N_{\partial, k}^q\big] \, \mathbb{P}(N = k) +
		\sum_{k = \lfloor 2 \lambda|A| \rfloor}^{\infty}( k^{-q} - (2 \lambda|A|)^{-q} )  \mathbb{E}\big[ N_{\partial, k}^q\big] \, \mathbb{P}(N = k)\\
		=(2 \lambda|A|)^{-q}  \mathbb{E}[N_{\partial}^q]  +
		\sum_{k = \lfloor 2 \lambda|A| \rfloor}^{\infty}( k^{-q} - (2 \lambda|A|)^{-q} )  \mathbb{E}\big[ N_{\partial, k}^q\big] \, \mathbb{P}(N = k)

	\end{EQA}
	with the absolute value of second sum being bounded using  the Cauchy-Schwarz inequality
	and large deviations by
	\begin{EQA}[c]
		c_1 \, \mathbb{E}[N_{\partial}^{2q}]^{1/2} \mathbb{P} (N \ge [2\lambda |A|])^{1/2} \le
		c_1 \, \mathbb{E}[N_{\partial}^{2q}]^{1/2} \exp(-c_2 n )\,,
	\end{EQA}
	for some constants \(c_1, c_2 >0 \). Thus, \eqref{EfronsIneqRev} follows.
\end{proof}
Proposition~\ref{EfronsProp} and Theorem~\ref{new_estimator_risk_wrap} immediately suggest the following bound for the remainder term in
the oracle inequality,
\begin{EQA}[c]
	r(\lambda,A) \le c   \mathbb{E}[|A \setminus \hat{A}|^4]^{1/4}\, \mathbb{P}(|A\setminus \hat{A}| \ge |A|/2)^{1/4}\,,
\end{EQA}
for some numeric constant \(c>0\). Therefore, the oracle inequality in Theorem~\ref{new_estimator_risk_wrap}
hinges upon only two probabilistic results:
\begin{itemize}
	\item the ratio \(\text{Var}(\abs{A\setminus\hat{A}})  / \mathbb{E}[ \abs{A\setminus\hat{A}}]\) of the moments of the missing volume,
	\item a uniform deviation inequality for the missing volume.
\end{itemize}
Both results are fairly involved and we shall only discuss here how to derive them for some simple classes of sets satisfying
Assumption~\ref{Assumption}.

\section{Classes of sets satisfying Assumption~\ref{Assumption}}
\label{ClassesOfBodies}
This section collects some examples of classes of sets that satisfy  Assumption~\ref{Assumption}. Note that
the class of all convex sets \({\mathbf C}_{\text{conv}}\) satisfies the assumption and was extensively studied in \cite{BaldinReiss16}.
The most involved statements
in the inference on convex sets were underpinned by the abundance of results from stochastic geometry on moment bounds
and deviation inequalities for the missing volume, see Lemma 4.6 in \cite{BaldinReiss16}. In particular,
the ratio \(\text{Var}(\abs{C\setminus\hat{C}})  / \mathbb{E}[ \abs{C\setminus\hat{C}}] \thicksim 1/\lambda\) is established in
\cite{pardon2011} for all convex sets \(C\)  in dimensions \(d = 1,2\). In dimensions \(d > 2\),
one can bound the ratio only for some subsets of the class of convex sets.
Thus, for a convex set \(C\)  with $C^2$-boundary of positive curvature, it is known thanks to \cite{reitzner2005central} that   \(\text{Var}(\abs{C\setminus\hat{C}} ) \lesssim \lambda^{-(d+3)/(d+1)}\). The lower bound
for the first moment, \( \mathbb{E}[ \abs{C\setminus\hat{C}}] \gtrsim \lambda^{-2/(d+1)}\), was shown in \cite{schutt1994random}.
For  a polytope \(C\),   the upper bound  \(\text{Var}(\abs{C\setminus\hat{C}} ) \lesssim \lambda^{-2}(\log \lambda)^{d-1}\) was obtained in \cite{barany2010variance},
while the lower bound for the first moment, \( \mathbb{E}[ \abs{C\setminus\hat{C}}] \gtrsim \lambda^{-1}(\log \lambda)^{d-1}\), was proved in \cite{barany1988convex}.
A uniform deviation inequality for convex sets
obtained in Thm.~1 in  \cite{Bru14c} allows to derive sharp upper bounds on the moments of the missing volume.
The proof of the deviation inequality exploited a bound on the entropy of convex sets.
It remains an intriguing open question in stochastic geometry whether \( \lambda \text{Var}(\abs{C\setminus\hat{C}}) \thicksim \mathbb{E}[ \abs{C\setminus\hat{C}}]  \)
holds universally for all convex sets in arbitrary dimensions.
Some of the classes of sets we consider here are much larger, yet very little has been known about them in the mathematical literature.
\subsection{$r$-convex sets}
\label{RConnSect}
We denote by \(B(x,r) \subset \mathbb{R}^d\)  (resp. \(B_\circ(x,r)\)) the closed (resp. open) ball with centre \(x\)  and radius \(r\).
\begin{definition}
	A compact set \(C_r\) in \({\mathbf E} \subset \mathbb{R}^d\) is called \emph{\(r\)-convex} for \(r > 0\), if its complement is the union
	of all open Euclidean balls of diameter \(r\) that are disjoint to \(C_r\), i.e. if
	\begin{EQA}[c]
		C_r = \bigcap_{B_\circ^c(x,r) \cap C_r = \varnothing} B_\circ^c(x,r)\,.
	\end{EQA}
	We denote the class of \(r\)-convex sets by \({\mathbf C}_r\)
\end{definition}
Note that an \(r\)-convex set fulfills the \emph{outside rolling ball condition}, i.e. for all \(y \in \partial C_r\) there is a closed ball \(B(x,r)\)
such that \(y \in \partial B(x,r)\) and \(B_\circ(x,r) \cap C_r = \varnothing\). Heuristically this means
that one can ``roll'' a ball of radius \(r\) freely over the boundary of a set. Note that according to the definition, \(r\)-convex
sets can have ``holes'' and do not need to be connected, see Figure~\ref{r-convex_alpha_varies} for some examples.
In the terminology of \cite{kendall1974foundations}, \(C_r \in {\mathbf C}_r \) means that the set \(C_r\) is \emph{trapped} by the balls of
radius \(r\).
The \(r\)-convex sets were introduced in
\cite{Perkal1956} and presumably independently in \cite{efimov1959some}; see \cite{walther1997,cuevas2012} and references therein for a recent work on estimation of \(r\)-convex sets.
In the literature, much more attention has been devoted to the sets satisfying the so-called \emph{inside and outside rolling ball condition}, when both
\(C_r\) and \(\overline{C_r^c}\) are \(r\)-convex, see \cite{mammen1995asymptotical,walther1996}. The reason probably is that sets with
smooth boundaries (with no angles) are sometimes easier to handle with geometric arguments, see \cite{lopez2008}.

The \({\mathbf C}_r\)-wrapping hull is defined by
\begin{EQA}[c]
	\hat{C}_r := \bigcap_{B_\circ^c(x,r) \cap \{X_1,...,X_N\} = \varnothing} B_\circ^c(x,r)
\end{EQA}
and often called the \emph{\(r\)-convex hull} in the literature.
Thus the oracle estimator in \eqref{OracleEst} has the following form
\begin{EQA}[c]
	\label{r_convex_hull_estimator}
	\hat\vartheta_{r,oracle} := \frac{N_\partial}{\lambda} + |\hat{C}_r|\,,
\end{EQA}
where \(N_\partial\) is the number of sample points lying on the the \(r\)-convex hull \(\hat{C}_r\). In order to investigate the performance
of this estimator according to Theorem~\ref{UpBoundOracle} it suffices to study \( \sup_{C_r \in {\mathbf C}_r}\mathbb{E}_{C_r}[|C_r\setminus \hat{C}_r|]\).
In fact the following result holds and it is a consequence of Theorem~\ref{UpBoundOracle}.
\begin{theorem}
	\label{ThmRconv}For known intensity $\lambda > 0$,
	the worst case mean squared error of the oracle estimator \(\hat\vartheta_{r,oracle}\) over the parameter class ${\mathbf C}_r$ decays as $\lambda\uparrow \infty$  like $\sup_{C_r \in {\mathbf C}_r}\mathbb{E}_{C_r}[|C_r\setminus \hat{C}_r|] / \lambda$ in dimension $d$:
	\begin{EQA}[c]
		\limsup_{\lambda\to \infty} \lambda \sup_{C_r \in {\mathbf C}_r, |C_r| >0} \Big\{\mathbb{E}\big[(\hat\vartheta_{r,oracle} -\abs{C_r})^2\big] /\mathbb{E}\big[|C_r\setminus \hat{C}_r|\big]\Big\}  < \infty\,.
	\end{EQA}
\end{theorem}
\begin{remark}
	Note that the class of convex sets \({\mathbf C}_{\text{conv}}\) belongs to \({\mathbf C}_r\) for all \(r>0\)
	and thus using Theorem 3.4 in \cite{BaldinReiss16} we have a lower bound on the rate of convergence,
	\begin{EQA}
		&&\inf_{ \hat\vartheta_\lambda}  \lambda^{(d+3)/(d+1)} \sup_{C_r \in {\mathbf C}_r}\mathbb{E}_{C_r } [ ( | C_r | -\hat\vartheta_\lambda)^2] \\
		&&\quad \quad   \ge
		\inf_{ \hat\vartheta_\lambda}  \lambda^{(d+3)/(d+1)} \sup_{C \in {\mathbf C}_{\text{conv}}}\mathbb{E}_{C } [ ( | C | -\hat\vartheta_\lambda)^2] > 0\,,
	\end{EQA}
	where the infimum extends over all estimators \(\hat\vartheta_\lambda\) in the PPP model with intensity \(\lambda\).
	Furthermore,  the rate $\lambda^{-(d+3)/(d+1)}$ is achieved up to a logarithmic factor for sets \(C_r \in {\mathbf C}_r \)
	with a smooth boundary following \cite{lopez2008}.
\end{remark}
Following Section~\ref{DataDriveEstimation}, the mean squared error of the estimator
\begin{EQA}[c]
	\label{FinEstRConv}
	\hat\vartheta_r := \frac{N + 1}{N_\circ + 1} |\hat{C}_r|\,,
\end{EQA}
satisfies the oracle inequality in Theorem~\ref{new_estimator_risk_wrap}. Upper bounding
the functions \(\alpha(\lambda, C_r)\) and \(r(\lambda,C_r)\) and
establishing an exact rate of convergence
of the risk of the estimator \(\hat\vartheta_r\)  requires sharp probabilistic
bounds similarly to the convex case and is beyond the scope of the present paper.
In order to compute \(\hat\vartheta_r\) in practice, it is crucial to know the radius \(r\).
It is clear that \({\mathbf C}_{r_2} \subset {\mathbf C}_{r_1}  \) for all \(r_2 > r_1 > 0\).
Define the true radius \(r^\star\) corresponding to a set \(C\) as
\begin{EQA}[c]
	r^\star = \sup \{r> 0: C \in {\mathbf C}_r \}\,.
\end{EQA}
It is intuitively evident that it is better to underestimate the radius \(r^\star\), because \({\mathbf C}_{r^\star} \subset {\mathbf C}_{r} \)
for all \(r^\star > r > 0\).
On the other hand, there is a typical bias-variance tradeoff in choosing the optimal parameter \(r\).
We prefer to use large values of \(r\) when the number of sample points is scarce,
and can afford to use small values  \(r < r^\star\) when there is an abundance of the sample points.
We here propose a procedure for estimating \(r^\star\) based on Lepski's method \cite{lepskii1992asymptotically},
see also a more accessible reference, Section 8.2.1 in \cite{giné2015mathematical}.
Fix some \(R \in \mathbb{R}_{+}\) and \(K \in {\mathbb N} \) and break the interval \((0,R)\) down into \(K + 1\) pieces \(0 < r_1 < ... < r_K < R\)
of equal length. Define an estimator for \(\hat{r}\) as
\begin{EQA}[c]
	\label{adaptR}
	\hat{r} := \inf \big\{ r_{k-1} |\, \exists   k^\prime \le k\, : \,  | \hat\vartheta_{r_k} -  \hat\vartheta_{r_{k^\prime}} | >  \varkappa_n \big\} \wedge r_K \,,
\end{EQA}
with \(\varkappa_n = N_\delta/ n^{2}\). This calibration is suggested by Theorem~\ref{UpBoundOracle} in view of Corollary~\ref{EfronsIdenPoisson}.
The asymptotic behaviour of the estimator \(\hat\vartheta_{\hat{r}}\) depends on an exact deviation inequality on the missing volume \(|C_r \setminus \hat{C}_r|\).
This question however is beyond the scope of the present paper. We provide a numerical study of this
adaptation procedure in Section~\ref{SimulationsSection}.

\subsection{Compact sets}
Interestingly the class of all compact sets \({\mathbf K}\) of non-zero Lebesgue measure satisfies Assumption~\ref{Assumption} as well.
The richness of this class makes it foremost for conducting statistical inference, yet
very little has been proposed and studied so far.
Estimation of this class of sets was studied in \cite{devroye1980detection}, where it was shown that
the union of small Euclidean balls centred at the points of the sample is a consistent estimator of a compact set.
The \({\mathbf K}\)-wrapping hull is just the union of sample points and so \(N_\partial = N\) and \(|\hat{K}| = 0\) a.s.
Hence for the oracle estimator in \eqref{OracleEst} we have
\begin{EQA}[c]
	\hat\vartheta_{{\mathbf K},oracle} := \frac{N}{\lambda}\,.
\end{EQA}
This estimator is unbiased
and from \eqref{RiskNaiveEst} the following result immediately follows.

\begin{lemma}For known intensity $\lambda > 0$,
	the worst case mean squared error of the oracle estimator \(\hat\vartheta_{{\mathbf K},oracle}\) over the parameter class ${\mathbf K}$ decays as $\lambda\uparrow \infty$  like $\lambda^{-1}$:
	\begin{EQA}[c]
		\sup_{K \in {\mathbf K}} \frac{1}{|K|} \mathbb{E} \big[(\hat\vartheta_{{\mathbf K},oracle} - |K|)^2\big]  =  \frac{1}{\lambda}\,.
	\end{EQA}
\end{lemma}
It seems impossible without imposing further structure on the class \({\mathbf K}\) to estimate \(\lambda\) in this scenario.

\subsection{Polytopes}
It was noted in \cite{BaldinReiss16}, that the estimator of the volume based on the
\({\mathbf C}_{\text{conv}}\)-wrapping hull estimator (the convex hull \(\hat{C}\)) is adaptive to the class of polytopes \({\mathbf P}\) (see Remark~3.3).
In fact, the estimator
\begin{EQA}[c]
	\label{polytopes_hull_estimator}
	\hat\vartheta_{{\mathbf P},oracle} := \frac{N_\partial}{\lambda} + |\hat{C}|\,
\end{EQA}
satisfies
\begin{lemma}
	\label{PolyLemR}For known intensity $\lambda > 0$,
	the worst case mean squared error of the oracle estimator \(\hat\vartheta_{{\mathbf P},oracle}\) over the parameter class ${\mathbf P}$ decays as $\lambda\uparrow \infty$  like $\lambda^{-2}(\log(\lambda))^{d-1}$:
	\begin{EQA}[c]
		\limsup_{\lambda\to \infty} \lambda^{2}(\log(\lambda))^{1-d} \sup_{P \in {\mathbf P}, |P| > 0} \Big\{\mathbb{E}\big[(\hat\vartheta_{{\mathbf P},oracle} -\abs{P})^2\big]\Big\}  < \infty\,.
	\end{EQA}
\end{lemma}
We stress here, however, that the class polytopes \({\mathbf P}\) does not satisfy Assumption~\ref{Assumption}. The framework
applies to the class of convex sets \({\mathbf C}\) and Lemma~\ref{PolyLemR} only allows to improve the rate for the subclass of \({\mathbf C}\).
The class \({\mathbf P}\) is stable only under finite intersections; taking arbitrary (possibly uncountable) intersections,
one can obtain an element not lying in the class.

\subsection{Polytopes with fixed directions of outer unit normal vectors}
\label{PolFixeDirSec}
The class of polytopes \({\mathbf P}_{{\mathbf S}_k} \) with fixed directions \({\mathbf S}_k = \{u_1,...,u_k\}\)
of outer unit normal vectors \(u_k \) belonging to the unit sphere \(\mathbb{S}^{d-1}\) provides another interesting example  of intersection stable sets. We assume the class is well-defined in the sense that there exists a polytope \(P_{{\mathbf S}_k}\) whose outer unit normal vectors  are  exactly \(\{u_1,...,u_k\}\). Without loss of generality we may assume \({\mathbf E} =P_{{\mathbf S}_k} \).
The \({\mathbf P}_{{\mathbf S}_k}\)-wrapping hull \(\hat{P} \) is a polytope with at most
\(k\) facets and is given by
\begin{EQA}[c]
	\hat{P} := \bigcap_{P  \in {\mathbf P}_{{\mathbf S}_k}: \{X_1,...,X_N\} \in P} P\,.
\end{EQA}
The oracle estimator and the data-driven estimator are thus defined as
\begin{EQA}[c]
	\label{polytopes_fixed}
	\hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} ,oracle} := \frac{N_\partial}{\lambda} + |\hat{P}|\,,
	\quad
	\hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} }  = \frac{N + 1}{N_\circ + 1} |\hat{P}|\,,
\end{EQA}
where the number of points lying on the boundary of the wrapping hull \(N_\partial\)
is equal to the number of facets of the wrapping hull and hence upper-bounded by \(k\).
According to the general scheme, the rate of convergence of the risk for the oracle estimator
\(	\hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} ,oracle} \)
and the final estimator  \(	\hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} } \) rests upon a deviation inequality for the missing volume and is established in the following theorem which is proved in the Appendix.

\begin{theorem}
	\label{polytopesFixed}
	The worst case mean squared error of the  estimator  \(	\hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} } \) over the parameter class ${\mathbf P}_{{\mathbf S}_k} $ satisfies:
	\begin{EQA}[c]
		\sup_{ P \in {\mathbf P}_{{\mathbf S}_k}} \mathbb{E}\big[( \hat\vartheta_{{\mathbf P}_{{\mathbf S}_k} } - |P|)^2 \big]
		\lesssim \frac{kW(\lambda/k) }{\lambda^2}\,,
	\end{EQA}
	whenever \(\lambda |P| \ge 1\), where \(W\) is the Lambert-W function that satisfies \(W(z{\rm e}^{z}) = z\). The rate can further
	be upper-bounded by \(k \log(\lambda / k )/\lambda^2\).
\end{theorem}

\section{Illustrative simulations}
\label{SimulationsSection}
In this section, we  illustrate the performance of the proposed estimators for a class of \(r\)-convex sets.
Our primal example of an \(r\)-convex set for simulations is the annulus \(C_{r^\star} = B(0.5,0.5)\setminus B(0.5,0.25)\). Thus clearly \(C_{r^\star} \in {\mathbf C}_r \) for all \(0 < r \le 0.25\). Figure~\ref{r-convex_alpha_varies} depicts the \(r\)-convex hull estimator \eqref{r_convex_hull_estimator} for \(r = 0.01, 0.04,0.2,1\) based on the observations of the PPP with \( \lambda  = 300\).
An important observation is that once the value of \(r\) is larger than the true radius \(r^\star\)
of an \(r\)-convex set, the \(r\)-convex hull essentially misses the ``holes'' of radius \(r^\star\). One should bear this in mind
when using large values of \(r\)  for constructing the oracle estimator when the number of observation points is small. This subtle issue is depicted in Figure~\ref{r_convex_fixe_alpha}, where
the root mean squared error of the oracle estimator \(\hat\vartheta_{r,oracle}\) for the volume  (black line), based on the \(r\)-convex hull with \(r = 0.04\), converges to the area of the ``hole'' of size \(\pi (0.25)^2 \approx  0.196\). Another striking point is that when the number of observations is small, the
\(r\)-convex hull with a small value of \(r\) essentially coincides with the points themselves and thus the RMSE of the oracle estimator
\(\hat\vartheta_{r,oracle}\) coincides with the RMSE of \(\hat\vartheta_{{\mathbf K},oracle}\) and equals \(|C_{r^\star}|/\lambda \) (red line in Figure~\ref{r_convex_fixe_alpha})! Finally we depict RMSE estimates for
the oracle estimator \(\hat\vartheta_{r,oracle}\) for different \(r\) in Figure~\ref{r_convex_n_varies_alpha_fixed}. One can clearly see the regions of decreasing
value of the RMSE, the fairly flat value of RMSE and the jump when \(r\) becomes larger than the true parameter \(r^\star\).
Table~\ref{table1} further collects the Monte Carlo estimates  of the number of points  \(N_\circ\) lying inside the wrapping hull,
the number of points \(N_\partial\)  on the boundary of the wrapping hull and the number of
isolated points \(N_{iso}\) of the boundary of the wrapping hull. For analyzing the performance
of the
adaptive estimator proposed in Section~\ref{RConnSect}, we break  the interval \([0.06, 0.5]\) into pieces of length \(0.02\),
compute the estimates of the radius \(\hat{r}\) from \eqref{adaptR} and the estimates of the RMSE of \(\hat\vartheta_{\hat{r}}\)
based on 200 Monte Carlo iterations in Table~\ref{table2}.

\begin{figure}[tp]
	\centering
	\includegraphics[width=0.8\linewidth]{r_varies_3.jpeg}\\[-5mm]
	\caption[]{The four weakly-convex set with \(r^\star = 0.25\) (blue), observations of the PPP with \(\lambda = 300\) (points)  and  their \(r\)-convex hulls for different values of \(r\) (black).}
	\label{r-convex_alpha_varies}
\end{figure}

\begin{figure}[tp]
	\centering
	\includegraphics[width=0.7\linewidth]{r_const_n_varies_3.jpeg}\\[-5mm]
	\caption[]{Monte Carlo RMSE estimates for the oracle estimator for the volume of the annulus \(B(0.5,0.5)\setminus B(0.5,0.25)\) with respect to the sample size.}
	\label{r_convex_fixe_alpha}
\end{figure}

\begin{figure}[tp]
	\centering
	\includegraphics[width=0.7\linewidth]{n_const_r_varies_3.jpeg}\\[-5mm]
	\caption[]{Monte Carlo RMSE estimates for the oracle estimator for the volume of the annulus \(B(0.5,0.5)\setminus B(0.5,0.25)\) with respect to \(r\).}
	\label{r_convex_n_varies_alpha_fixed}
\end{figure}

\begin{table}[tp]
	\footnotesize
	\resizebox{\linewidth}{!}{%
		\begin{tabular}{c  c  c  c  c c c
			\multicolumn{7}{c}{\(r = 0.04\) }\\
			\(n = \lambda / |A|\)  & \(N_\circ\)& \(N_\partial\) & \(N_{iso}\) & \( RMSE(\hat\vartheta_{r,oracle})\) & \( RMSE(\hat\vartheta_{r})\) &
			\( \frac{RMSE(\hat\vartheta_{r}) }{RMSE(\hat\vartheta_{r,oracle})} \)   \\
			50 & 0.13 & 49.8 & 44 &0.087 &0.55 &  6.31 \\
			100 & 1.5 &  98.37& 66 &0.059& 0.33 & 4.65 \\
			200 & 23.6 &  175.8& 58 & 0.042& 0.15 & 3.7 \\
			300 & 90.4 &  211 & 33 &0.039 &0.109 & 2.82 \\
			400 & 191.6 & 210 & 17 & 0.043 &0.085 &  1.97\\
	\end{tabular}}\vspace{10pt}
	\resizebox{\linewidth}{!}{
		\begin{tabular}{c  c  c  c  c c c }
			\multicolumn{7}{c}{\(r = 0.1\) }\\
			\(n = \lambda / |A|\)  & \(N_\circ\)& \(N_\partial\) & \(N_{iso}\) & \( RMSE(\hat\vartheta_{r,oracle})\) & \( RMSE(\hat\vartheta_{r})\) &
			\(  \frac{RMSE(\hat\vartheta_{r}) }{RMSE(\hat\vartheta_{r,oracle})} \)  \\
			50 & 8.5 & 42.16 &8.52   & 0.071 &0.138 & 1.94\\
			100 & 45.9 & 53.34 & 1.37 & 0.043 &0.064&1.48  \\
			200 &  138.2 & 60.95 &0.03  &0.021  &0.027& 1.26 \\
			300 &  233.4 & 68.08 &0  &  0.015&  0.018&1.20\\
			400 & 326.1 &  74.40& 0& 0.013 & 0.015 & 1.15\\
	\end{tabular}}\vspace{10pt}
	\resizebox{\linewidth}{!}{
		\begin{tabular}{c  c  c  c  c c c  }
			\multicolumn{7}{c}{\(r = 0.25\) }\\
			\(n = \lambda / |A|\)  & \(N_\circ\)& \(N_\partial\) & \(N_{iso}\) & \( RMSE(\hat\vartheta_{r,oracle})\) & \( RMSE(\hat\vartheta_{r})\) &
			\(  \frac{RMSE(\hat\vartheta_{r}) }{RMSE(\hat\vartheta_{r,oracle})} \)   \\
			50 & 24.75 & 24.21 & 0.06 & 0.061 &0.085 &1.39 \\
			100 &68.58  & 29.60 & 0 &0.033  & 0.0405& 1.20\\
			200 & 163.75 &36.03  &  0&0.018  & 0.019 &1.04\\
			300 & 261.44 & 40.68 &  0&0.0108  & 0.0124 &1.13\\
			400 & 357.41 &  44.17&  0&  0.0096 &  0.0104 &1.076\\
	\end{tabular}}\vspace{10pt}
	\resizebox{\linewidth}{!}{
		\begin{tabular}{c  c  c  c  c c c  }
			\multicolumn{7}{c}{\(r = 0.3\) }\\
			\(n = \lambda / |A|\)  & \(N_\circ\)& \(N_\partial\) & \(N_{iso}\) & \( RMSE(\hat\vartheta_{r,oracle})\) & \( RMSE(\hat\vartheta_{r})\) &
			\( \frac{RMSE(\hat\vartheta_{r}) }{RMSE(\hat\vartheta_{r,oracle})}  \)  \\
			50 &  30.71 &18.70  &  0& 0.208 &0.340  &1.628\\
			100 &77.59  &23.26  & 0 & 0.2002 & 0.258&1.29 \\
			200 &  170.30& 29.39 & 0 & 0.1982 &0.232 &1.17 \\
			300 & 265.17 & 33.89 & 0 & 0.1978 & 0.223& 1.13 \\
			400 &362.43  &  37.89& 0 &0.1987 &0.219 &  1.10\\
	\end{tabular}}

	\caption{Monte Carlo RMSE estimates for the oracle estimator \(\hat\vartheta_{r,oracle}\) and for the fully data-driven
		estimator \(\hat\vartheta_r\) for the volume of the annulus \(A = B(0.5,0.5)\setminus B(0.5,0.25)\)
		with respect to \(r\) and \(n = \lambda |A|\), the number of points lying inside the wrapping hull \(N_\circ\),
		the number of points on the boundary of the wrapping hull \(N_\partial\) and the number of
		isolated points of the boundary of the wrapping hull \(N_{iso}\).}
	\label{table1}
\end{table}

\begin{table}[tp]
	\footnotesize

	\begin{center}
		\begin{tabular}{c  c  c
			\(n = \lambda / |A|\)  & \(\hat{r} \) & \( {RMSE(\hat\vartheta_{\hat{r}})}  \) \\
			50 & 0.088 & 0.36  \\
			100 & 0.085 &  0.160 \\
			200 & 0.084 &  0.069 \\
			300 & 0.105 &  0.033  \\
			400 & 0.125 & 0.0182 \\
			500 & 0.149 & 0.0123 \\
			1000 & 0.165 & 0.0056 \\
		\end{tabular}
	\end{center}
	\caption{Monte Carlo RMSE estimates for the adaptive estimator \(\hat\vartheta_{\hat{r}}\)
		for the volume of the annulus \(A = B(0.5,0.5)\setminus B(0.5,0.25)\)
		with respect to \(n = \lambda |A|\).}
	\label{table2}
\end{table}

\subsection{Appendix}
\subsubsection{Proof of Theorem~\ref{SuffComplThm}}
Sufficiency follows from the Neyman factorisation criterion applied to the likelihood function \eqref{fPPeb},
while completeness follows by definition provided that we show
\begin{EQA}[c]
	\forall A \in {\mathbf A}: \mathbb{E}_A\bigl[T(\hat{A})\bigr] = 0 \implies  T(\hat{A}) = 0 \;\;\; \mathbb{P}_{\mathbf E}-a.s.
\end{EQA}
for any \(\mathcal{A}\)-measurable function \(T: {\mathbf A} \to \mathbb{R}\).
From the likelihood  in \eqref{fPPeb} for $\lambda=\lambda_0$, we derive
\begin{EQA}
	\mathbb{E}_A\bigl[T(\hat{A})\bigr] & = &\mathbb{E}_{\mathbf E}\Bigl[T(\hat{A}) \exp\bigl(\lambda\abs{{\mathbf E}\setminus A}\bigl){\bf 1}(\hat{A} \subseteq A )\Bigr] \,.
\end{EQA}
Since \(\exp(\lambda\abs{{\mathbf E}\setminus A})\) is deterministic,  we have \(\forall A \in {\mathbf A}\
\begin{EQA}[c]
	\mathbb{E}_A\bigl[T(\hat{A})\bigr] = 0 \implies
	\mathbb{E}_{\mathbf E}\bigl[T(\hat{A}){\bf 1}(\hat{A} \subseteq A) \bigr] = 0\,. \,
\end{EQA}
Splitting \(T = T^{+} - T^{-}\)
with non-negative \(\mathcal{A}\)-measurable functions \(T^{+}\) and \(T^{-}\), we infer that the measures
\(\mu^\pm(B)=\mathbb{E}_{\mathbf E}[T^{\pm}(\hat{A}){\bf 1}(\hat{A} \in B)] \), $B\in \mathcal{A}$, agree on \(\{[B] \,|\, B \in {\mathbf A} \}\), where
\([B] = \{A\in {\mathbf A} |  A \subseteq B \}\). Since the brackets \(\{[B] \,|\, B \in {\mathbf A} \}\) generate the \(\sigma\)-algebra \(\mathcal{A}\) the measures
\(\mu^\pm(B)\) agree on all sets in \(\mathcal{A}\), in particular on  \(\{T > 0\} \) and \(\{T < 0\} \), which entails $\mathbb{E}_{\mathbf E}[T^+(\hat{A})]=\mathbb{E}_{\mathbf E}[T^-(\hat{A})]=0$.
Thus, \(T(\hat{A}) = 0\) holds  \(\mathbb{P}_{\mathbf E}\)-a.s.

\subsubsection{Proof of Theorem~\ref{polytopesFixed}}
Let us denote by \(\rho_1(A,B) = |A\triangle B|\)  the symmetric distance between
two compact subsets \(A\) and \(B\) of the compact convex set \({\mathbf E}\) in \(\mathbb{R}^d\).
Recall that an \emph{\(\epsilon\)-net} of the class \({\mathbf P}_{{\mathbf S}_k}\) with respect to the metric \(\rho_1\) is a collection \(\{P^1,...,P^{N_\epsilon} \} \in {\mathbf P}_{{\mathbf S}_k }\)
such that for each \(P \in {\mathbf P}_{{\mathbf S}_k}\), there exists \(i \in \{1,...,N_\epsilon\} \) such that \(\rho_1(P,P^i) \le \epsilon\). The \emph{\(\epsilon\)-covering
	number} \(N({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon)\) is the cardinality of the smallest \(\epsilon\)-net. The \emph{\(\epsilon\)-entropy} of the class \({\mathbf P}_{{\mathbf S}_k}\) is
defined by \(H({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon) = \log_2 N({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon)\). Furthermore, it follows by dilation of a set that for  \(\hat{P} \in {\mathbf P}_{{\mathbf S}_k}\) there exists \(\hat{m} \in \{1,...,N_\epsilon\} \) such that   \(\hat{P} \subseteq P^{\hat{m}} \subseteq P  \) and \(\rho_1(\hat{P}, P^{\hat{m}}) \le c \epsilon\) for some universal constant \(c > 1\) and \( \epsilon\) small enough. We thus obtain for all \(P \in {\mathbf P}_{{\mathbf S}_k}\) and \(x > 0\),
\begin{EQA}
	\mathbb{P}\big(|P \setminus \hat{P}| > x/\lambda + 2 c \epsilon \big) & \le& 	\mathbb{P}\big(|P \setminus P^{\hat{m}} | > x/\lambda + c \epsilon \big)
	\\
	&\le&  \sum_{m: |P \setminus P^{m} | > x/\lambda +  c \epsilon }
	\mathbb{P}\big(\mathcal{N} (P \setminus P^{m}) = 0 \big)  \\
	&\le& \exp\big( -x - c\lambda \epsilon + H({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon)  \big) = {\rm e}^{-x}\,,
\end{EQA}
plugging in \(\epsilon\)
that solves \(H({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon) = c \lambda \epsilon \). The \(\epsilon\)-covering
number \(N({\mathbf P}_{{\mathbf S}_k}, \rho_1, \epsilon)\)  of the class \({\mathbf P}_{{\mathbf S}_k}\) can  be bounded by \((C/\epsilon)^{k}\) for some universal constant \(C > 1\). As a result,
the asymptotic rate follows using  Fubini's theorem combined with Theorem~\ref{UpBoundOracle} and Theorem~\ref{new_estimator_risk_wrap}.

\begin{comment}
\subsubsection{Proof of Theorem~\ref{MinimaxLowerBoundThm}}
The upper bound in \eqref{eq:ddvvvbe} follows directly from Theorem~\ref{UpBoundOracle}.
For the lower bound, we reduce the minimax risk to the Bayes risk.
For the prior on \({\mathbf A}\), define the random
variables \(\xi_k \overset{i.i.d.}{\sim} \text{Bernoulli}(p) \), i.e. \(\mathbb{P}(\xi_k = 0) = p\), \(\mathbb{P}(\xi_k = 1)=  1 - p\) and set \(A  = A^\circ \setminus \cup_{k = 1}^{M} \xi_k C_k \), where \(A^\circ\) is a  set in the class \({\mathbf A}\)  of the biggest  volume and \(C_k\) are the disjoint pieces of volume \(\epsilon\) such
that \(A \in {\mathbf A}\) for all possible values of \(\xi_k\). Clearly, \(M \ge 1\)  because of the intersection stability property.  Thus, we derive
\begin{EQA}[c]
\inf_{\hat{\vartheta}_\lambda}\sup_{A \in {\mathbf A} } \mathbb{E}_A [ ( | A | -\hat{\vartheta}_\lambda)^2] \ge
\inf_{\hat{\vartheta}_\lambda}\sup_{A \in {\mathbf A} } \mathbb{E}_\xi \mathbb{E}_{A(\xi)}  [ ( | A| -\hat{\vartheta}_\lambda)^2] \ge
\mathbb{E}_\xi \mathbb{E}_{A(\xi)}  [ ( | A| -\hat{\vartheta}_B)^2]\,,
\end{EQA}
where \(\hat{\vartheta}_B\) is a Bayes-optimal estimator satisfying
\begin{EQA}[c]
\hat\vartheta_B = \mathbb{E}\bigl[|A| \, \big| \, (X_1, ..., X_N)\bigr] =
|A^\circ| - \sum_{k = 1}^{M} |C_k| \mathbb{E}\bigl[\xi_k \, \big| \, (X_1, ..., X_N)\bigr]
\end{EQA}

Using the Bayes formula and  the likelihood function
\eqref{fPPeb}, we derive
\begin{EQA}
\hat{\xi}_k &\stackrel{\operatorname{def}}{=}&  \mathbb{E}\bigl[{\xi}_k \, \big| \, (X_1, ..., X_N) \bigr]  = \mathbb{P}\bigl({\xi}_k = 1\, \big| \,  (X_1, ..., X_N)  \bigr) \\
& = & \frac{  (d \mathbb{P}_{\bar{g}_k} /d \mathbb{P}_{0})(d \mathbb{P}_{\bar{g}_0} / d \mathbb{P}_{0})^{-1} p  }{ 1 - p + (d \mathbb{P}_{\bar{g}_k} /d \mathbb{P}_{0})(d \mathbb{P}_{\bar{g}_0} / d \mathbb{P}_{0})^{-1} p }\\
& = &\frac{p{\rm e}^{\lambda\int_{I_k} \bar{g}_k dx} }{1 - p + p{\rm e}^{\lambda\int_{I_k} \bar{g}_k dx} } {\bf 1}\Big(\forall X_i^{(1:d-1)} \in I_k:  X_i^{(d)} \ge \bar{g}_k(X_i^{(1:d-1)})\Big),
\end{EQA}
where the superscript in \(X_i^{(1:d-1)} \) and \( X_i^{(d)}\) denotes the corresponding components of the vector \(X_i\),
\(\bar{g}_k(x) = g_k(x) - \|x\|^2\), \(\bar{g}_0(x) = 0\) and   \( \mathbb{P}_{\bar{g}_k} \), \(\mathbb{P}_{\bar{g}_0}\)
stand for the measures associated with the convex sets whose boundaries are \(\bar{g}_k(x)\) and \(\bar{g}_0(x)\)
in the k-th cube \(I_k\).
Observing
$\hat\theta_B =    \sum_{k = 1}^{M} \hat{\xi}_k \int_{[0,1]^{d-1}} g_k(x) dx  - \int_{[0,1]^{d-1}} \| x \|^2$
and  \(\text{Var}\bigl({\xi}_k \, \big| \,  (X_1, ..., X_N)  \bigr) =  \hat{\xi}_k ( 1- \hat{\xi}_k)\), the Bayes risk can be calculated as
\begin{EQA}
\mathbb{E}_\xi \mathbb{E}_{C(\xi)}  [ ( | C| -\hat{\theta}_B)^2] & = & \sum_{k = 1}^{M} \text{Var}( \tilde{\xi}_k  - \xi_k) \Bigl(\int_{I_k} g_k(x) dx\Bigr)^2 \\
& = & \sum_{k = 1}^{M} \mathbb{E}\Bigl[ \text{Var}\bigl({\xi}_k \, \big| \,  (X_1, ..., X_N)  \bigr) \Bigr] \Bigl(\int_{I_k} g_k(x) dx\Bigr)^2 \\
& = & \sum_{k = 1}^{M}  \frac{p (1 - p) }{1 - p + p{\rm e}^{\lambda\int_{I_k} \bar{g}_k(x)dx} }  \Bigl(\int_{I_k} g_k(x) dx\Bigr)^2 \,.
\end{EQA}
Note, that \(\lambda\int_{I_k}  \bar{g}_k(x) dx \le \lambda\int_{I_k}  g_k(x) dx  \le \lambda b_1 h_\lambda^{d + 1}  \le b_2\) for some  constants  \(b_1, b_2>0\) if we choose \( h_\lambda \thicksim \lambda^{-1/( d + 1)} \).
This implies  the bound for  \(b_3, b_4,b_5>0\)
\begin{EQA}
\mathbb{E}_\xi \mathbb{E}_{C(\xi)}  [ ( | C| -\hat{\theta}_B)^2]  & \ge & M b_3 \bigl(\int_{I_k} g_k(x) dx\bigr)^2 \\
& \ge & M b_3 h_\lambda^4 a^2 \bigl(\int_{I_k} u(\frac{x-q_k}{h_\lambda}) dx\bigr)^2
\ge    b_4  h_\lambda^{d+3} \ge  b_5 \lambda^{\frac{d+3}{d+1}}\,,
\end{EQA}
which completes the proof.
\end{comment}

\begin{comment}
\subsubsection{Proof of Theorem~\ref{UpperBoundEntropy}}
We first prove a uniform deviation inequality on for the wrapping hull as a set estimator
for the set itself.  It appears easier to prove in the uniform model with the fixed
number of points distributed uniformly over a set. We exploit a flexible entropy approach
developed in \cite{mammen1995asymptotical} for a similar set estimation problem.

\begin{theorem}
Let \( \epsilon_n\) satisfy \(\epsilon_n = H_{\epsilon_n} / n := H({\mathbf A}, \rho_1, \epsilon) /n\). Then in the uniform model, there exist \(c  > 0\) such that the following uniform deviation inequality holds
\begin{EQA}[c]
\label{DevIneq}
\sup_{A \in {\mathbf A}} \mathbb{P}\Big(n\big(\abs{\hat{A} \setminus A} - 2 \epsilon_n \big) \ge x\Big) \le c{\rm e}^{-x/2}\,,\quad \forall x > 0\,.
\end{EQA}
\end{theorem}
\begin{proof}
Let \({\mathbf A} \) be any class satisfying Assumption~\ref{Assumption} and
choose a packing
\(\epsilon\)-net \(\{A_1,...,A_{N_{\epsilon}} \}\) of cardinality \(N_\epsilon = N({\mathbf A},\rho_1, \epsilon)\).
Let \(\hat{A}_m  :=\operatornamewithlimits{argmin}_{ A \in \{A_1,...,A_{N_{\epsilon}\} }} \rho_1(\hat{A}, A) \). Using the triangle inequality, we thus have \(\forall y > 0\)
\begin{EQA}[c]
\mathbb{P}_A\bigl(\rho_1(A, \hat{A}) \ge y \bigr) \le \mathbb{P}_A\bigl(\rho_1(A, \hat{A}_m) \ge y -\epsilon \bigr)\,.
\label{eq:0}
\end{EQA}
where we have used that

the event \(\rho_1(A, \hat{A}_m) \ge y -\epsilon  \) implies
that there exists collection of sets in the packing set required to removed from

Removing a set from the packing set decreases the volume of the union of all sets in the packing set by \(\epsilon\).
\end{proof}

The following corollary is a simple consequence of Fubini's theorem.
\begin{corollary}
The minimax convergence rate for estimating \(A \in {\mathbf A}\) with the wrapping hull \(\hat{A} \)
is parametrised by the metric entropy of an underlying class as follows:
\begin{EQA}[c]
\sup_{A \in {\mathbf A}, |A| >0} \mathbb{E}\big[\abs{A \setminus \hat{A}}^q\big]  \lesssim \epsilon_n^q \,.
\end{EQA}
for any integer \(q\).
\end{corollary}
The same rate of convergence holds in the PPP model using Poissonization replacing \( n\) with \(\lambda \). Furthermore, we immediately derive an upper bound on the risk of
the oracle estimator \( \hv_{\text{oracle}}\) for volume estimation
\begin{corollary}
In the PPP model, the mean squared risk of the oracle estimator satisfies
\begin{EQA}[c]
\sup_{A \in {\mathbf A}, |A| >0} \mathbb{E} \big[(\hv_{\text{oracle}}- |A|)^2\big] \lesssim \epsilon_\lambda/ \lambda \, .
\end{EQA}
\end{corollary}

\end{comment}

\begin{comment}
\subsubsection{Set-indexed processes, stopping sets and the proof of Theorem~\ref{LemMeasCond}}
Recall that a random compact set \(\mathcal{ K}\) is called an \(\mathcal{F}_K\)-stopping set if \(\{\mathcal{ K} \subseteq K\} \in \mathcal{F}_K\)
for all \(K \in {\mathbf K}\). The sigma-algebra of \(\mathcal{ K}\)-history is defined as
$\mathcal{F}_{\mathcal{ K}} = \{A \in \mathfrak{F}: A \cap \{ \mathcal{ K} \subseteq K \} \in \mathcal{F}_K\,\,\, \forall K \in {\mathbf K} \}$, where
\(\mathfrak{F} =   \sigma(\mathcal{F}_K; K \in {\mathbf K})\). Let us collect some useful results proved in \cite{BaldinReiss16}.
The following statement is a generalisation of the analogous statement for time-indexed stochastic processes.

\begin{lemma}
\label{LemMeas}
The random variable \(\mathcal{N}({\mathcal{ K}})\) is measurable with respect to \(\mathcal{F}_\mathcal{ K}\) for any stopping set \(\mathcal{ K}\).
\end{lemma}
The set-indexed process \((\mathcal{N}(K), K \in {\mathbf K})\) has independent increments, i.e. for
\(K_1,\ldots,K_m \in {\mathbf K}\) with \(K_i \subseteq K_{i+1}\), $i=1,\ldots,m-1$, the random variables \(\mathcal{N}(K_{i+1}) - \mathcal{N}(K_i)=\mathcal{N}(K_{i+1}\setminus K_i)\)
are independent (by the independence of the PPP on disjoint sets).
Furthermore, we say that a set-indexed, \((\mathcal{F}_K)\)-adapted integrable process
\((X_K, K \in {\mathbf K})\) is a martingale if  \(\mathbb{E}[X_B| \mathcal{F}_A] = X_A\) holds for any \(A, B \in {\mathbf K}\) with \(A \subseteq B\).
By the independence of increments, the process
\begin{EQA}[c]
M_K \stackrel{\operatorname{def}}{=} \mathcal{N}(K) - \lambda|K|,\quad K\in{\mathbf K}\,,
\end{EQA}
is clearly a martingale with respect to its natural filtration \((\mathcal{F}_K,K\in{\mathbf K})\).
Applying the martingale characterisation  of the set-indexed Poisson process,
see Thm. 3.1 in \cite{ivanoff1994martingale} (analogue of  Watanabe's characterisation for the Poisson process), to
the martingale \((M_K, K \in {\mathbf K})\), we show in Proposition~\ref{psm} that the process \(\mathcal{N}\) is even a \emph{strong Markov}
set-indexed process, see \cite{BaldinReiss16} for an accurate proof.
\begin{proposition}
\label{psm}
The set-indexed process \((\mathcal{N}(K), K\in{\mathbf K})\) is \emph{strong Markov}  at every stopping set \(\mathcal{ K}\). More precisely,
conditionally on \(\mathcal{F}_{\mathcal{ K}}\) the process \((\mathcal{N}(K\setminus \mathcal{ K}),K\in{\mathbf K})\) is  a Poisson point process with intensity $\lambda$ on $\mathcal{ K}^c$.
In particular, \(\mathcal{N}(K\setminus \mathcal{ K} ) \, \big| \, \mathcal{F}_{\mathcal{ K}}  \sim \text{Poiss}(\lambda|K\setminus \mathcal{ K}|)\) holds for all  \(K \in {\mathbf K}\) and
\begin{EQA}[c]
\label{NumPointsMissVol}
\mathbb{E}[\mathcal{N}(K\setminus \mathcal{ K} )] = \lambda \mathbb{E}[ |K\setminus \mathcal{ K}|]\,.
\end{EQA}
\end{proposition}
In particular,
Proposition~\ref{psm} yields \eqref{NccCs} using
that the closed complement \(\mathcal{ K} = \overline{\hat{A}^c}\) of the wrapping hull is a stopping set.
Moreover, \eqref{NumPointsMissVol} yields a rather striking relation between the number of points \(N_\partial\) lying on the wrapping hull \(\hat{A}\) and
the missing volume \(|A \setminus \hat{A}|\), which resembles Efron's identity:
\begin{EQA}[c]
\label{efronsIdent2}
\mathbb{E}[N_\partial] = \lambda\, \mathbb{E}[ |A\setminus \hat{A}|]\,.
\end{EQA}
The last statement of Theorem~\ref{LemMeasCond},
that  \(\mathcal{F}_{\hat\mathcal{ K}} = \sigma(\hat{A}) \) for \(\mathcal{ K} = \overline{\hat{A}^c}\) follows from the following lemma.
It can be seen as a generalisation of the
interesting fact that
for a time-indexed Poisson process the sigma-algebra \(\sigma(\tau)\) associated
with the first jump time \(\tau\) coincides with the sigma-algebra
of \(\tau\)-history \(\mathcal{F}_\tau\).

\begin{lemma} The sigma-algebra \(\sigma(\hat{A}) =  \sigma(\{\hat{A} \subseteq B, B \in \mathcal{A}\})\)
coincides with  the sigma-algebra \( \mathcal{F}_{\hat\mathcal{ K}}\) of \(\hat\mathcal{ K}\)-history for the stopping set \(\mathcal{ K} = \overline{\hat{A}^c}\), i.e. \(\sigma(\hat{A}) = \mathcal{F}_{\hat\mathcal{ K}}\).
\end{lemma}
The main idea of the proof is that the sigma-algebra \( \mathcal{F}_{\hat\mathcal{ K}}\)  coincides with the sigma-algebra
\(\sigma(\{\mathcal{N}(\hat\mathcal{ K}\cap K), K \in {\mathbf K}\})\) generated by the  process stopped at $\hat\mathcal{ K}$. This statement
can be derived from Thm. 6, Ch. 1 in \cite{shiryaev2007optimal}.
Finally, observe that by definition of the wrapping hull $\mathcal{N}(\overline{\hat{A}^c}\cap K)=\mathcal{N}((\partial\hat{A})\cap K)$.
Modulo null sets, $\mathcal{N}((\partial\hat{A})\cap K)$ counts the number of vertices of $\hat{A}$ in $K$ and is thus $\sigma(\hat{A})$-measurable.

\end{comment}

\section*{Acknowledgements}
The author is  grateful to Markus Rei{s^\star} at
Humboldt-Universit{\"a}t zu Berlin for hosting him in
January 2016 and  valuable discussions which originated this research. The author further thanks Quentin Berthet for helpful comments.

\bibliographystyle{economet}

redpajama_set_name: RedPajamaArXiv


text:
The Alaska Senate is about to roll out its proposed budget, with spending cuts deeper than the House's
Author: James Brooks
The leaders of the Senate Finance Committee speak with legislative staff Thursday morning, April 18, 2019. At left is Sen. Bert Stedman, R-Sitka and co-chair of the committee. Seated second from right is Sen. Natasha von Imhof, R-Anchorage and the other co-chair. (James Brooks / ADN)
JUNEAU — Members of the Alaska Senate's finance committee are scheduled Monday to unveil a state budget proposal that rejects extensive cuts proposed by Gov. Mike Dunleavy but sticks closer to the governor than a separate alternative approved by the Alaska House of Representatives earlier this month.
"Everything's on track for the operating budget: On track, on calendar, so far, so good," said Sen. Bert Stedman, R-Sitka and co-chairman of the Senate Finance Committee, in a conversation with reporters Thursday morning. "We've been working on the writing of that bill for several days, and we've got some more loose ends to tie up."
"I would expect the Senate operating budget to come in — on our agencies — lower than the House," he added.
Three plans on the table
If it were graphed, the state's budget process would look like two turkey wishbones laid end to end, starting and ending at a single point. The governor introduces his proposal as a single starting point, then the House and Senate take separate tracks. Each has subcommittees to set budgets for individual state departments. Each uses its finance committee to combine those department budgets into a proposal, then amend that proposal. Each chamber votes on that proposal on the floor. The House and Senate have reached a budget compromise and come back to a single point.
The House has completed its proposal and is waiting on the Senate to do the same. (The House Finance Committee has turned its attention to crime-fighting legislation in the meantime.) According to a schedule provided by Sen. Natasha von Imhof, R-Anchorage and a co-chair of the Senate Finance Committee, that committee will examine the Senate's alternative budget next week and consider amendments to the alternative.
A schedule released this week calls for the finance committee to begin meeting Monday afternoon on the alternative budget, with an analysis from the nonpartisan legislative finance division on Tuesday. After a Wednesday break to allow for amendments to be drafted, the committee will return to the budget for the last two days of the week.
As soon as the first few days of May, the proposal will be on the Senate floor and approved by senators. That would leave two weeks for a compromise before the end of the regular session. The governor's veto power would come into play after that, if he chooses.
Differences and similarities
Though the Senate plan is shaping up to be closer to the governor's than the House's, the Senate is siding with the House on budget issues that would require additional legislation. For example, Dunleavy proposed taking more than $420 million in petroleum and fisheries taxes from local communities for the state treasury. That would require legislative permission, and von Imhof said last week that it will not happen.
The governor has proposed eliminating the state's senior benefits program. A Senate subcommittee rejected that elimination, following the lead of the House.
The House and Senate have agreed to keep legislation from last year that funds the state's K-12 public schools next year without any cuts. The governor has challenged the Legislature's ability to keep that appropriation.
Within the House budget is a plan to pay $70 million in oil and gas tax credits from the accounts of the Alaska Industrial Development and Export Authority. Stedman said his preference is that the Senate pay that amount — plus more — from the state treasury instead.
"I don't like that strategy using AIDEA receipts … I'm not saying we won't do it," he said.
Paying oil companies from the treasury would inflate the Senate budget figures above the House, he said, even if the Senate cuts more money from agencies.
"That is one of the big components we're still talking about," he said.
He said he would also prefer some "backfill" into the corpus of the Alaska Permanent Fund, but details of that idea were not immediately clear.
Spending from savings
All three proposals — House, Senate and the governor's — reject the idea of raising taxes to erase the state deficit or preserve a larger Permanent Fund dividend. Instead, each has adopted a binary stance: Cutting state services raises the dividend, and reducing the dividend allows more services.
The budgets proposed by the governor and House erase the deficit without spending from the Constitutional Budget Reserve. Permanent Fund spending is limited to the sustainable level approved last year. (The governor's budget allows spending from the Statutory Budget Reserve if planned Medicaid savings fall short. The House doesn't use it.)
On Thursday, Stedman didn't rule out the possibility of the Senate budget spending from savings.
"If you don't have recurring revenue, you have to go into your savings accounts to make your payroll. You don't have any choice," he said. "So we're going to have that discussion."
Dividends on the table
Dunleavy's budget proposal is built around the idea of a "full" dividend paid using the statutory formula in state law. That would be about $3,000, plus or minus $300, according to the calculations of Ed King, chief economist with the Office of Management and Budget. The House budget proposal doesn't include a dividend amount. Speaking on the House floor last week, lawmakers said their intention is to address the dividend in separate legislation, but Rep. Neal Foster, D-Nome, estimated the House dividend at $1,284 during an April 12 press conference if lawmakers used only the money left over after state services are paid for.
Von Imhof told reporters last week that the Senate intends to include the dividend as a set line item in the budget, and Stedman said Thursday that the budget will "most likely" include a dividend amount. That amount may be close to the House figure or slightly above.
He said that with the governor's figure and the Senate's expected level of budget cuts, "clearly there'd be a deficit on recurring revenue with a $3,000 dividend."
Health care cuts
One of the biggest differences among the three budgets is in the way they treat health care spending. The House approved a $3.2 billion budget for the Department of Health and Social Services. The governor proposed $2.49 billion, though the department has since conceded that not all of those savings are possible this year. The Senate's figure is $3.14 billion, according to legislative finance division numbers.
"We cut over $135 million in (undesignated general fund) dollars. The budget that we are proposing is below $1 billion UGF for the first time since 2011," von Imhof said, referring to the amount of direct state support for the department, which is funded primarily with federal money.
Medicaid spending in the Senate budget is $25 million below the House's figure.
University spending above House budget
The House and Senate budget plans each reject the governor's proposal to reduce state support for the university system by $134 million, or 40 percent. The House has settled on a $10 million cut, but the proposal from the Senate's university budget subcommittee is even less than that: $5 million. Thanks to a budgeting change created by the Senate subcommittee, that reduction would come from the university's Fairbanks and Anchorage campuses only.
Ferry system funding
Stedman, who represents a coastal community, oversaw the budget discussions for the Department of Transportation, but the Alaska Marine Highway System didn't escape cuts under his management. The draft Senate proposal cuts ferry funding by $44 million, according to figures from the transportation subcommittee. The governor's proposal involved $98 million in cuts and no service after Oct. 1. The House's proposal is for $10.9 million in cuts, according to figures from the nonpartisan finance division.
Stedman said he doesn't look at the Senate figure as a cut, since the governor proposed no service.
"That is economically dangerous and unworkable for coastal Alaska," he said of the governor's plan.
"Clearly, reduced service is better than no service at all," Stedman said.
James Brooks covers state government, the Alaska Legislature and general assignments for the Daily News. He previously reported and edited for the Juneau Empire, Kodiak Daily Mirror and Fairbanks Daily News-Miner.

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