Containment and inscribed simplices

TL;DR

This paper explores conditions under which convex sets contain translates of polytopes and their projections, establishing equivalences and counterexamples related to containment and projection properties in Euclidean space.

Contribution

It introduces new equivalence conditions for convex set containment involving polytopes and projections, and constructs examples where projections do not imply full set containment.

Findings

Equivalence between polytope containment and set containment for convex bodies.

Projection conditions are equivalent to polytope containment for certain dimensions.

Existence of convex sets with prescribed projection properties but lacking full containment.

Abstract

Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W. It is then shown that, if K is a compact convex set in R^n having at least d+2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K, while L does not contain a translate of K. In particular, such a convex body L exists whenever dim(K) > d.