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 ...

Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Mi...

As Schneider [50] observes, the classical Brunn-Minkowski theory had its origin at the turn of the 19th into the 20th century, when Minkowski joined a method of combining convex bodies (which became known as Minkowski addition) with that of ordinary volume. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is that of projection body (precise definitions to fol...

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...

– 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 ...

According to the MathSciNet search engine, approximately 100 papers and books pertaining to the Brunn-Minkowski inequality have appeared since Das Gupta's 1980 article Brunn-Minkowski and its aftermath 56]. An attempt is made here to present an up-to-date survey.