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Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ containe …

I am interested in the $2$-Wasserstein distance for probabilities over ${\mathbb R}^n$, $$W_2(\mu,\nu)=\left(\inf\int_{{\mathbb R}^n\times{\mathbb R}^n}|w-v|^2\pi(v,w)\right)^{1/2}$$ where the infimum …

Homogenization is a process that assigns to a positive definite-valued map $x\mapsto S(x)$ a non-trivial but physically meaningful average $\bar S$. There are various settings, for instance stochastic …

Let me recall the Brunn-Minkowski inequality, which states concavity of ${\rm vol}^{1/d}$ for domains in ${\mathbb R}^d$: $${\rm vol}(A+B)^{1/d}\ge{\rm vol}(A)^{1/d}+{\rm vol}(B)^{1/d},$$ with equalit …