On polynomially integrable convex bodies

TL;DR

This paper proves that in odd dimensions, the only smooth convex bodies with polynomial parallel section functions of degree at least n-1 are ellipsoids, highlighting a unique geometric property in odd-dimensional spaces.

Contribution

It establishes a characterization of polynomially integrable convex bodies in odd dimensions, showing they must be ellipsoids when the degree is at least n-1.

Findings

In odd dimensions, polynomially integrable convex bodies are ellipsoids for degree N ≥ n-1.

Such bodies do not exist in even dimensions or for N < n-1 in odd dimensions.

The result contrasts with previous findings in even dimensions and lower degrees.

Abstract

An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky.