Some Orlicz-Brunn-Minkowski type inequalities for (dual) quermassintegrals of polar bodies and star dual have been introduced. In this paper, we generalize the results establish some mixed bodies.

The purpose of this work is to give a direct proof the multiplicative Brunn-Minkowski inequality in nilpotent Lie groups based on Hadwiger-Ohmann's one classical Euclidean space.

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.

In this paper, we construct an injection A × B → M ×M from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If A and B are disjoint, our construction allows to inject two copies of A × B into M ×M . These injections imply a positively curved Brunn-Minkowski inequ...

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...

In his Annals of Mathematics paper [2], Berndtsson proves an important result on the Nakano positivity holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting L2-functions with respect to a family weight functions {e−φ(t,⋅)}t∈U, varying in t∈U⊂Cm, over pseudoconvex domain. Using variant Hörmander's theorem due Donnelly and Fefferman, we show that Berndtsson's holds u...