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 ...

Theorem 7.1 (Brunn-Minkowski). If A, B ⊆ R n satisfy some mild assumptions (in particular, convexity suffices), then [vol (A + B)] 1 n ≥ [vol (A)] 1 n + [vol (B)] 1 n where A + B = { a + b : a ∈ A and b ∈ B}.

Starting from a mass transportation proof of the Brunn-Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincaré-type trace inequality on convex sets that is also proved in sharp form.

This year marks the 150(th) anniversary of the presentation by Gregor Mendel of his studies of plant hybridization to the Brunn Natural History Society. Their nature and meaning have been discussed many times. However, on this occasion, we reflect on the scientific enterprise and the perception of new discoveries.

Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of po...

– Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton–Jacobi equations. By the infimum-convolution description of the Hamilton–Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobo...

Here we collect some notation and basic lemmas used throughout this note. Throughout, for a random variable X, ‖X‖p denotes (E |X|). It is known that ‖ · ‖p is a norm for any p ≥ 1 (Minkowski’s inequality). It is also known ‖X‖p ≤ ‖X‖q whenever p ≤ q. Henceforth, whenever we discuss ‖ · ‖p, we will assume p ≥ 1. Lemma 1 (Khintchine inequality). For any p ≥ 1, x ∈ R, and (σi) independent Rademac...