An asymmetric convex body with maximal sections of constant volume

TL;DR

This paper constructs an asymmetric convex body in higher dimensions where all maximal hyperplane sections have equal volume, answering a longstanding question in convex geometry.

Contribution

It demonstrates the existence of asymmetric convex bodies with uniform maximal section volumes in all dimensions greater than two, resolving a question posed by V. Klee in 1969.

Findings

Existence of asymmetric convex bodies with constant maximal section volume in all dimensions > 2

Negative answer to Klee's 1969 question

Advances understanding of convex body geometry

Abstract

We show that in all dimensions d>2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.