On the diminishing process of B. Tóth

Abstract

Let $K$ and $K_0$ be convex bodies in ${\mathbb{R}}^d$, such that $K$ contains the origin, and define the process $(K_n,p_n)$, $n\ge 0$, as follows: let $p_{n+1}$ be a uniform random point in $K_n$, and set $K_{n+1}=K_n\cap (p_{n+1}+K)$. Clearly, $(K_n)$ is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in ${\mathbb{R}}^d$. We study this process for $K$ being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.

29 pages, 10 figures