On the measure of the one-skeleton of the sum of convex compact sets

Abstract

AbstractFor any two compact convex sets in a Euclidean space, the relation between the volume of the sum of the two sets and the volume of each of them is given by the Brünn-Minkowski inequality. In this note we prove an analogous relation for the one-dimensional Hausdorff measure of the one-skeleton of the above sets. Also, some counterexamples are given which show that the above results are the best possible in some special cases.