Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1. As observed by Schneider [21], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R, with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classica...

Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms based on Schuster?s homomorphisms. In 2018, Feng and He gave the concept of (p,q)-mixed geominimal surface area according to Lutwak, Yang Zhang?s volume. this article, associated with areas homomorphisms, we establish some inequalities including two Brunn-Minkowski type inequalities, a cyclic inequality monotonic inequalities.

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube {0, 1}N . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm, and Villani could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, includ...

It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y )/ √ 2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of th...

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube {0, 1} . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic natu...

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of R n. For the existence part we apply the variational method introduced by Jerison in: Adv. Math. 122 (1996), pp. 262–279. Uniqueness follows from the Brunn–Minkowski inequality for the torsional rigidity and corresponding ...