  The convex body chasing problem, introduced by Friedman and Linial, is a
competitive analysis problem on any normed vector space. In convex body
chasing, for each timestep $t\in\mathbb N$, a convex body $K_t\subseteq \mathbb
R^d$ is given as a request, and the player picks a point $x_t\in K_t$. The
player aims to ensure that the total distance $\sum_{t=0}^{T-1}||x_t-x_{t+1}||$
is within a bounded ratio of the smallest possible offline solution.
  In this work, we consider the nested version of the problem, in which the
sequence $(K_t)$ must be decreasing. For Euclidean spaces, we consider a
memoryless algorithm which moves to the so-called Steiner point, and show that
in a certain sense it is exactly optimal among memoryless algorithms. For
general finite dimensional normed spaces, we combine the Steiner point and our
recent previous algorithm to obtain a new algorithm which is nearly optimal for
all $\ell^p_d$ spaces with $p\geq 1$, closing a polynomial gap.

Data Structures and Algorithms;Metric Geometry