On the diminishing process of B. T\'oth

TL;DR

This paper investigates a stochastic process involving nested convex bodies in Euclidean space, analyzing its convergence behavior for specific shapes and extending some results to one-dimensional cases with non-uniform distributions.

Contribution

It introduces a new process of convex body intersection with random points, analyzing its diminishing behavior for regular shapes and providing novel one-dimensional results.

Findings

Convergence of the nested convex bodies to a limit shape.

Behavior characterized for simplices, cubes, and polygons with odd vertices.

New results obtained for one-dimensional non-uniform distributions.

Abstract

Let $K$ and $K_0$ be convex bodies in $\mathbb{R}^d$, such that $K$ contains the origin, and define the process $(K_n, p_n)$, $n \geq 0$, 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.