On bodies in $\mathbb{R}^5$ with directly congruent projections or sections

TL;DR

This paper proves that in five-dimensional space, convex bodies with certain congruence conditions on their projections or sections are essentially identical or negatives of each other, under symmetry restrictions.

Contribution

It establishes a uniqueness result for convex bodies in $\,\mathbb{R}^5$ based on congruence of projections or sections, extending prior geometric symmetry results.

Findings

Convex bodies with congruent projections on certain subspaces are identical up to sign and translation.

Projections with no rotational symmetries lead to uniqueness of the bodies.

Results also apply to sections of star bodies, broadening the scope of geometric congruence.

Abstract

Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many diameters, such that their projections onto all $4$ dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections have no rotational symmetries, and the projections of $K,L$ on certain 3 dimensional subspaces have no symmetries, then $K=\pm L$ up to a translation. We also prove the corresponding result for sections of star bodies.