On separably integrable symmetric convex bodies

An infinitely smooth symmetric convex body $K\subset {\mathbb{R}}^d$ is called $k$-separably integrable, $1\le k<d$, if its $k$-dimensional isotropic volume function $V_{K,H}(t)={\mathcal{H}}^d(\{\mathit{x}\in K:\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\mathit{x},H^{\perp })\le t\})$ can be written as a finite sum of products in which the dependence on $H\in \mathrm{G}\mathrm{r}(k,{\mathbb{R}}^d)$ 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.