On the hollow enclosed by convex sets

TL;DR

This paper investigates the geometric structure of hollows formed by convex sets in Euclidean space, proving that the closure of their convex hull is always a simplex, thus revealing a fundamental property of such configurations.

Contribution

It establishes that the convex hull of a hollow enclosed by convex sets in R^d is always a d-simplex, extending understanding of convex set intersections.

Findings

The convex hull of a hollow is a d-simplex.

Any n-critical family with n ≤ d has a hollow enclosed by a simplex.

The result generalizes properties of convex set intersections in Euclidean spaces.

Abstract

For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If $n=d$ then a lemma on the intersection of convex sets due to Klee implies that the $d+1$ members of the $d$-critical family enclose a `hollow' in $R^d$, a bounded connected component of $R^d\setminus\bigcup_{i=0}^n C_i.$ Here we prove that the closure of the convex hull of a hollow in $R^d$ is a $d$-simplex.