Let ?n be the standard Gaussian measure on Rn. We prove that for every symmetric convex sets K,L in Rn and ??(0,1),?n(?K+(1??)L)1n???n(K)1n+(1??)?n(L)1n, thus settling a problem raised by Gardner Zvavitch (2010). This is analogue of classical Brunn–Minkowski inequality Lebesgue measure. also show that, fixed ??(0,1), equality attained if only K=L.

In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, Cap <mm...

We introduce f -divergence, a concept from information theory and statistics, for convex bodies in R. We prove that f -divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the Lp affine surface area from the Lp Brunn Minkowski theory are special cases of f -divergences.