Abstract. For metric measure spaces verifying the reduced curvature-dimension condition CD∗(K,N) we prove a series of sharp functional inequalities under the additional assumption of essentially nonbranching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfy...

We introduce f -divergence, a concept from information theory and statistics, for convex bodies in R. We prove that f -divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the Lp affine surface area from the Lp Brunn Minkowski theory are special cases of f -divergences.

This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.

Let ρ and λ be Banach function norms with the Fatou property. Then the generalized Minkowski integral inequality ρ(λ(f x)) ≤ M λ(ρ(f y)) holds for all measurable functions f (x, y) and some fixed constant M if and only if there exists 1 ≤ p ≤ ∞ such that λ is p–concave and ρ is p–convex.