It is well-known that the Hölder-Rogers inequality implies the Minkowski inequality. Infantozzi [6] observed implicitely and Royden [15] proved explicitely that the reverse implication is also true. In this note we discuss and give a new proof of this perhaps surprising fact. Mathematics subject classification (2000): 26D15.

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman’s reverse Brunn-M...

We prove an inequality of the form γn ≥ Cn(γn−1) between Hermite’s constants γn and γn−1. This inequality yields also a new proof of the Minkowski-Hlawka bound ∆n ≥ ζ(n)21−n for the maximal density ∆n of n−dimensional lattice-packings. 1

Birandom variable is a generalization of random variable, which is defined as a measurable function from a probability space to the set of random variables. In order to further explore the mathematical properties of birandom variable, this paper first investigates several inequalities for birandom variables based on the chance measure and expected value operator, which are analogous to Hölder’s...

The classical Minkowski inequality in the Euclidean space provides a lower bound on total mean curvature of hypersurface terms surface area, which is optimal round spheres. In this paper we employ locally constrained inverse flow to prove properly defined analogue Lorentzian de Sitter space.