On bodies with directly congruent projections and sections

TL;DR

This paper investigates conditions under which convex bodies in four-dimensional space are congruent based on their projections, establishing that under certain symmetry and diameter conditions, bodies with directly congruent projections are essentially identical.

Contribution

The paper proves that convex bodies with directly congruent projections are congruent up to translation and orthogonal transformation under specific diameter and symmetry conditions, extending to star bodies and higher dimensions.

Findings

Convex bodies with directly congruent projections are congruent under certain conditions.

Results extend to sections of star bodies and higher-dimensional spaces.

Provides new criteria for congruence based on projections and symmetries.

Abstract

Let $K$ and $L$ be two convex bodies in ${\mathbb R^4}$, such that their projections onto all $3$-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfy an additional condition and some projections do not have certain symmetries, then $K$ and $L$ coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the $n$-dimensional versions of these results.