We investigate the validity and stability of various Minkowski-like inequalities for C 1 -perturbations ball. Let K ? R n be a domain (possibly not convex mean-convex) which is -close to prove sharp geometric inequality ( ? ? ? II d H ? ) 2 ? Per , where constant that yields equality when = B (and sum absolute values eigenvalues second fundamental form ). Moreover, any ? > 0 if sufficiently bal...

Abstract. A generalization of Young’s inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. ...

According to the MathSciNet search engine, approximately 100 papers and books pertaining to the Brunn-Minkowski inequality have appeared since Das Gupta's 1980 article Brunn-Minkowski and its aftermath 56]. An attempt is made here to present an up-to-date survey.

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...