On an inequality of Minkowski for mixed volumes (Q583612)

For convex bodies K and L of volume 1 and centroid 0 in \({\mathbb{R}}^ n\), the author proves the inequality \[ (*)\quad V(K,...,K,L)\geq 1+(\gamma_ n/nD^{n+1})h(K,L)^{n+1}. \] Here the left side is a mixed volume with (n-1) times the argument K, h(K,L) is the Hausdorff distance of K and L, D denotes the maximum of the diameters of K and L, and \(\gamma_ n\) is an explicitly given constant. The famous Minkowski inequality for mixed volumes says that V(K,...,K,L)\(\geq 1\), with equality only if \(K=L\) (under the assumptions on volumes and centroids as above). Thus (*) improves this inequality, giving an explicit estimate for the Hausdorff distance of K and L in terms of the deficit \(V(K,...,K,L)-1.\) The proof uses a similarly improved version of the Brunn-Minkowski theorem which the author had established previously [Geom. Dedicata 27, 357-371 (1988; Zbl 0652.52009)]. As special cases, stability versions for the isoperimetric and for Urysohn's inequality are obtained.