Corresponding to each origin-symmetric convex (or more general) subset of Euclidean n-space R, there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace ofR. This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the ...

We strengthen some known stability results from the Brunn-Minkowski theory and obtain new results of similar types. These results concern pairs of convex bodies for which either surface area measures, or counterparts of such measures in the Brunn-Minkowski-Firey theory, or geometrically significant transforms of such measures, are close to each other. MSC 2000: 52A20, 52A40.

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, an...

1.1 Brunn-Minkowski inequality 1.1 Theorem. (Brunn-Minkowski, ’88) If A and B are non-empty compact sets then for all λ ∈ [0, 1] we have vol ((1− λ)A+ λB) ≥ (1− λ)(volA) + λ(volB). (B-M) Note that if either A = ∅ orB = ∅, this inequality does not hold since (1−λ)A+λB = ∅. We can use the homogenity of volume to rewrite Brunn-Minkowski inequality in the form vol (A+B) ≥ (volA) + (volB). (1.1) We ...

The Brunn-Miknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This classical inequality in convex geometry was inspired by issues around the isoperimetric problem and was considered for a long time to belong to geometry, where its significance is widely recognized. However, it is by now clear that the Brunn-Miknowski ineq...

In [F] Firey extended the notion of the Minkowski sum, and introduced, for each real p, a new linear combination of convex bodies, what he called p-sums. E. Lutwak [Lu2], [Lu3] showed that these Firey sums lead to a Brunn-Minkowski theory for each p ≥ 1. He introduced the notions of p-mixed volume, p-surface area measure, and proved an integral representation and inequalities for p−mixed volume...

– 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 Brunn-Minkowski inequality theory plays an important role in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. It has many useful applications in combinatorics, stochastic geometry, and mathematical economics. In recent years, several authors including Ball [1, 2, 3], Bourgain and Lindenstrauss...

A dual capacitary Brunn-Minkowski inequality is established for the (n − 1)capacity of radial sums of star bodies in R. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in R, 1 ≤ p < n, proved by Borell, Colesanti, and Salani. When n ≥ 3, the dual capacitary BrunnMinkowski inequality follows from an inequality of...

Abstract. A generalization of Young’s inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. ...