We outline existence of the Brenier map. As an application we present simple proofs of the multiplicative form of the Brunn–Minkowski inequality and the Marton–Talagrand inequality. 1.

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...

Abstract This paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and equivalent Minkowski mixed log-volume.

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...

We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex.

with equality if and only if f and g are proportional. For p <0, we assume that f(x), g (x) >0. An (almost) improvement of Minkowski’s inequality, for p Î R\{0}, is obtained in the following Theorem: Theorem 1.2 Let f(x), g(x) ≥ 0 and p >0, or f(x), g(x) >0 and p <0. Let s, t Î R\{0}, and s ≠ t. Then (i) Let p, s, t Î R be different, such that s, t >1 and (s t)/(p t) >1. Then ∫ (f (x)+g(x))pdx ...

The above norm induces a metric d where d(f, g) = ‖f − g‖p. Note that d(f, g) = 0 if and only if f = g a.e. μ, in which case we identify f with g. The Lp norm, like all worthy norms, satisfies the triangle inequality: ‖f + g‖p ≤ ‖f‖p + ‖g‖p ; this is precisely Minkowski’s inequality. For random variables X, Y defined on the same probability space and having finite p’th moments, Minkowski’s ineq...

Moreover, we introduce a curvature-dimension condition CD(K, N) being more restrictive than the curvature bound Curv(M,d, m) ≥ K. For Riemannian manifolds, CD(K, N) is equivalent to RicM (ξ, ξ) ≥ K · |ξ|2 and dim(M) ≤ N . Condition CD(K,N) implies sharp version of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows ...

This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. The Brunn-Minkowski Theorem asserts that, for compact convex sets K,L ⊆ Rn, the n-th root of the Euclidean volume Vn is concave with respect to Minkowski combinations; that is, for λ ∈ [0, 1], Vn((1− λ)K + λL) ≥ (1− λ)Vn(K) + λVn(L). The equality condition asserts that if K and L both have positive ...