On separably integrable symmetric convex bodies

An infinitely smooth symmetric convex body K⊂Rd is called k-separably integrable, 1≤k<d, if its k-dimensional isotropic volume function VK,H(t)=Hd({x∈K:dist(x,H⊥)≤t}) can be written as a finite sum of products in which the dependence on H∈Grk(Rd) and t∈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.