An extension of Milman's reverse Brunn-Minkowski inequality

compact convex sets, followed by an analytical proof by Minkowski [Min]. The inequality (1)for compact sets, not necessarily convex, was first proved by Lusternik [Lu]. A very simple proof ofit can be found in [Pi 1], Ch. 1.It is easy to see that one cannot expect the reverse inequality to hold at all, even if it is perturbedby a fixed constant and we restrict ourselves to balls (i.e. convex symmetric compact sets with theorigin as an interior point). Take for instance A