Counterexamples Related to Rotations of Shadows of Convex Bodies

TL;DR

This paper constructs specific convex body examples demonstrating that projections can be rotated to fit within another body, but the bodies themselves cannot, revealing limitations in projection-based containment assumptions.

Contribution

It provides explicit counterexamples and necessary conditions related to rotations of convex bodies and their projections in high-dimensional spaces.

Findings

Counterexamples where projections can be rotated into each other but bodies cannot

Necessary conditions for body containment based on projections

Insights into the geometric structure of convex bodies and their shadows

Abstract

We construct examples of two convex bodies $K,L$ in $\mathbb{R}^n$, such that every projection of $K$ onto a $(n-1)$-dimensional subspace can be rotated to be contained in the corresponding projection of $L$, but $K$ itself cannot be rotated to be contained in $L$. We also find necessary conditions on $K$ and $L$ to ensure that $K$ can be rotated to be contained in $L$ if all the $(n-1)$-dimensional projections have this property.