The slices of a cone and a characterization of ellipsoids

Let C be a convex cone in R-d with non-empty interior and a compact basis K . If H-1 and H-2 are any two parallel hyperplanes tangent to K, whose slice s with C are two other compact basis K-1 and K-2, let D, D-1 and D-2 be the truncated subcones of C generated by K, K-1 and K-2.(.) We prove that K is an ellipsoid if, and only if, vol (D)(2) = vol (D-1) vol (D-2) for every s uch pair of hyperplanes H-1 and H-2.