On bodies with congruent sections or projections

TL;DR

This paper constructs examples of convex bodies in higher dimensions that have congruent projections onto all subspaces but are not congruent themselves, disproving a long-standing conjecture.

Contribution

It provides the first counterexamples to Nakajima and Süss's conjecture, showing that congruent projections do not imply overall congruence of convex bodies.

Findings

Counterexamples in dimensions n ≥ 3

Disproof of Nakajima and Süss's conjecture

Convex bodies with congruent projections but non-congruent bodies

Abstract

In this paper, we construct two convex bodies $K$ and $L$ in $\mathbb{R}^n$, $n\geq 3$, such that their projections $K|H$, $L|H$ onto every subspace $H$ are congruent, but nevertheless, $K$ and $L$ do not coincide up to a translation or a reflection in the origin. This gives a negative answer to an old conjecture posed by Nakajima and S\"uss.