On separably integrable symmetric convex bodies

TL;DR

This paper classifies symmetric convex bodies with a special volume function property, showing they are ellipsoids or balls depending on the dimension, thus extending previous classifications of polynomially integrable bodies.

Contribution

It provides a complete classification of $k$-separably integrable symmetric convex bodies, identifying them as ellipsoids or Euclidean balls based on the parity of $d-k$.

Findings

If $d-k$ is even, $K$ is an ellipsoid.

If $d-k$ is odd, $K$ is a Euclidean ball.

The classification extends previous results on polynomially integrable bodies.

Abstract

An infinitely smooth symmetric convex body $K\subset\mathbb R^d$ is called $k$-separably integrable, $1\leq k<d$, if its $k$-dimensional isotropic volume function $V_{K,H}(t)=\mathcal H^d(\{\boldsymbol x\in K:\mathrm{dist}(\boldsymbol x,H^\perp)\leq t\})$ can be written as a finite sum of products in which the dependence on $H\in\mathrm{Gr}(k,\mathbb R^d)$ and $t\in\mathbb R$ is separated. In this paper, we will obtain a complete classification of such bodies. Namely, we will prove that if $d-k$ is even, then $K$ is an ellipsoid, and if $d-k$ is odd, then $K$ is a Euclidean ball. This generalizes the recent classification of polynomially integrable convex bodies in the symmetric case.