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arXiv · 2306.17127

On separably integrable symmetric convex bodies

Vladyslav Yaskin·Bartłomiej Zawalski
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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.

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Vladyslav Yaskin, Bartłomiej Zawalski. 2023-06-29. On separably integrable symmetric convex bodies. https://arxiv.org/abs/2306.17127

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