On separably integrable symmetric convex bodies

An infinitely smooth symmetric convex body K C R-d is called k-separably integrable, 1 <= k < d, if its k-dimensional isotropic volume function V-K,V-H(t) = H-d ({X is an element of K : dist (X, H-perpendicular to ) <= t}) can be written as a finite sum of products in which the dependence on H is an element of Gr(k)(R-d) and t is an element of 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).