If you can hide behind it, can you hide inside it?

TL;DR

This paper investigates the geometric properties of convex sets related to their projections and decomposability, establishing conditions under which certain convex bodies are reliable based on their boundary normals and decomposability.

Contribution

It characterizes d-reliability of convex sets in terms of their decomposability and boundary normals, providing new criteria for reliability and decomposability in convex geometry.

Findings

d-decomposability implies d-reliability

Smooth convex bodies are not d-reliable

1-reliability iff 1-decomposability (parallelotope)

Abstract

Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of a compact convex set K, it must follow that L contains a translate of K. It is shown that, for 1 <= d <= n-1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in R^n is d-reliable if and only if, for all m >= d+2, no m unit normals to regular boundary points of L form the outer unit normals of a (m-1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in R^n is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in R^n is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable.