A problem of Klee on inner section functions of convex bodies

TL;DR

This paper constructs examples of convex bodies that are not uniquely determined by their inner section functions, answering a longstanding question negatively and showing that different bodies can share the same maximal section areas.

Contribution

It provides explicit constructions of convex bodies with identical inner section functions, including non-symmetric and symmetric bodies, disproving the uniqueness conjecture.

Findings

Constructed pairs of convex bodies with identical inner section functions

Bodies can be arbitrarily close to the unit ball

Counterexamples include both symmetric and non-symmetric bodies

Abstract

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.