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.

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedde...

This paper is devoted to similarity and symmetry measures for convex shapes whose deenition is based on Minkowski addition and the Brunn-Minkowski inequality. This means in particular that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they ca...