  We study the question of how to compute a point in the convex hull of an
input set $S$ of $n$ points in ${\mathbb R}^d$ in a differentially private
manner. This question, which is trivial non-privately, turns out to be quite
deep when imposing differential privacy. In particular, it is known that the
input points must reside on a fixed finite subset $G\subseteq{\mathbb R}^d$,
and furthermore, the size of $S$ must grow with the size of $G$. Previous works
focused on understanding how $n$ needs to grow with $|G|$, and showed that
$n=O\left(d^{2.5}\cdot8^{\log^*|G|}\right)$ suffices (so $n$ does not have to
grow significantly with $|G|$). However, the available constructions exhibit
running time at least $|G|^{d^2}$, where typically $|G|=X^d$ for some (large)
discretization parameter $X$, so the running time is in fact $\Omega(X^{d^3})$.
  In this paper we give a differentially private algorithm that runs in
$O(n^d)$ time, assuming that $n=\Omega(d^4\log X)$. To get this result we study
and exploit some structural properties of the Tukey levels (the regions $D_{\ge
k}$ consisting of points whose Tukey depth is at least $k$, for $k=0,1,...$).
In particular, we derive lower bounds on their volumes for point sets $S$ in
general position, and develop a rather subtle mechanism for handling point sets
$S$ in degenerate position (where the deep Tukey regions have zero volume). A
naive approach to the construction of the Tukey regions requires $n^{O(d^2)}$
time. To reduce the cost to $O(n^d)$, we use an approximation scheme for
estimating the volumes of the Tukey regions (within their affine spans in case
of degeneracy), and for sampling a point from such a region, a scheme that is
based on the volume estimation framework of Lov\'asz and Vempala (FOCS 2003)
and of Cousins and Vempala (STOC 2015). Making this framework differentially
private raises a set of technical challenges that we address.

Data Structures and Algorithms;Computational Geometry;Cryptography and Security;Machine Learning