β1 ψ dx− ab+ α1β1. Young’s inequality [2, 1, 4] states that for every a ∈ [α1, α2] and b ∈ [β1, β2] (2) 0 ≤ F (a, b), where the equality holds iff φ(a) = b (or, equivalently, ψ(b) = a). Among the classical inequalities Young’s inequality is probably the most intuitive. Indeed, its meaning can be easily grasped once the integrals are regarded as areas below and on the left of the graph of φ (see...

Jensen’s inequality is sometimes called the king of inequalities [4] because it implies at once the main part of the other classical inequalities (e.g. those by Hölder, Minkowski, Young, and the AGM inequality, etc.). Therefore, it is worth studying it thoroughly and refine it from different points of view. There are numerous refinements of Jensen’s inequality, see e.g. [3-5] and the references...

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the proce...

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed and surface area measures of bodies. We study generalizations these concepts to Borel with density Rn– particular, weighted versions volumes (the so-called measures) when dealing up three distinct then formulate analyze classical measures, obtain a new integral formula for measure As an application, w...

We prove a concentration inequality for the `q norm on the `p sphere for p, q > 0. This inequality, which generalizes results of Schechtman and Zinn, is used to study the distance between the cone measure and surface measure on the sphere of `p . In particular, we obtain a significant strengthening of the inequality derived in [NR], and calculate the precise dependence of the constants that app...

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 isoperimetric inequalities imply in a very general measuremetric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition...

The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets.

We derive whole series of new integral inequalities of the Hardy-type, with non-conjugate exponents. First, we prove and discuss two equivalent general inequa-li-ties of such type, as well as their corresponding reverse inequalities. General results are then applied to special Hardy-type kernel and power weights. Also, some estimates of weight functions and constant factors are obtained. ...

We present an alternative, short proof of a recent discrete version the Brunn–Minkowski inequality due to Lehec and second named author. Our also yields four functions theorem Ahlswede Daykin as well some new variants.