A description of continuous rigid motion compatible Minkowski valuations is established. As an application we present a Brunn–Minkowski type inequality for intrinsic volumes of these valuations.

– We present a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (noncommutative) probability theory. The proof relies on matrix approximation as used recently by P. Biane and F. Hiai, D. Petz and Y. Ueda to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered ...

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski’s inequality, a generalized isoperimetric inequality for mixed volumes that is typically pro...

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is th...

We prove a functional version of the Brunn-Minkowski inequality for restricted sums obtained by Szarek and Voicu-lescu. We only consider Lebesgue-measurable subsets of R n , and for A ⊂ R n , we denote its volume by |A|. If A, B ⊂ R n , their Minkowski sum is defined by A + B = {x + y, (x, y) ∈ A × B}. The classical Brunn-Minkowski inequality provides a lower bound for its volume. In their stud...

Week 1 (9/7/2010) . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Results and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Existence of Haar Measure . . . . . . . . . . . . . . . . . . . . . . . . . ...

In this paper, we first introduce a new concept of dual quermassintegral sum function of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov– Fenchel inequality and the Brunn–Minkowski inequality for mixed intersection...

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6],[1]) has an analogue for the free entropy χ(X...

Minkowski type inequalities for the seminormed fuzzy integrals on abstract spaces are studied in a rather general form. Also related inequalities to Minkowski type inequality for the seminormed fuzzy integrals on abstract spaces are studied. Several examples are given to illustrate the validity of theorems. Some results on Chebyshev and Minkowski type inequalities are obtained.

We derive a Brunn-Minkowski-type inequality regarding the volume of the Minkowski sum of degenerate sets, namely, line sets. Let A 1 : : : A n be one dimensional sets of unit length, and v