Non-uniqueness of convex bodies with prescribed volumes of sections and projections

TL;DR

This paper demonstrates that in four or more dimensions, there exist distinct convex bodies sharing identical maximal, central section, and projection volumes in every direction, highlighting non-uniqueness in geometric reconstruction.

Contribution

It proves the existence of non-unique convex bodies with identical section and projection volumes in higher dimensions, revealing limitations in geometric determination.

Findings

Existence of non-unique convex bodies in even dimensions ≥ 4

Convex bodies with identical section and projection volumes

Highlights non-uniqueness in geometric inverse problems

Abstract

We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.