A FAMILY of inequalities concerning inner products of vectors and functions began with Cauchy. The extensions and generalizations later led to the inequalities of Schwarz, Minkowski and Holder. The well known Holder inequality involves the inner product of vectors measured by Minkowski norms. In this paper, another step of extension is taken so that a Holder type inequality may apply to general...

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinal...

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.