In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality μ(λA+ (1− λ)B) ≥ λμ(A) + (1− λ)μ(B) holds true for an unconditional product measure μ with decreasing density and a pair of unconditional convex bodies A,B ⊂ R. We also show that the above inequality is true for any unconditional logconcave me...

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.

Here we collect some notation and basic lemmas used throughout this note. Throughout, for a random variable X, ‖X‖p denotes (E |X|). It is known that ‖ · ‖p is a norm for any p ≥ 1 (Minkowski’s inequality). It is also known ‖X‖p ≤ ‖X‖q whenever p ≤ q. Henceforth, whenever we discuss ‖ · ‖p, we will assume p ≥ 1. Lemma 1 (Khintchine inequality). For any p ≥ 1, x ∈ R, and (σi) independent Rademac...

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.

The Brunn–Minkowski inequality gives a lower bound of the Lebesgue measure of a sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory of convex bodies. This topic has many interactions with isoperimetry or functional analysis. Our aim here is to report some recent aspects of these interactions involving optimal mass transport or the Heat equation. A...

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the Lp analogue of the classical BusemannPetty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corr...

Relative to the Gaussian measure on R, entropy and Fisher information are famously related via Gross’ logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by Stam. We establish a dimension-free inequality that interpolates among these relations. Several interesting corollaries follow: (i) the deficit in the LSI satisfies a conv...

The Brunn-Minkowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This inequality plays a crucial role in the theory of convex bodies and has many interactions with isoperimetry and functional analysis. Stability of optimizers of this inequality in one dimension is a consequence of classical results in additive combinatorics....