4 problems

Let $\lambda$ denote the uniform probability measure on $[0,1]$, let $\Pi(\lambda,\lambda)$ be the set of Borel probability measures on $[0,1]^2$ whose two marginals are $\lambda$,…

Vector-valued Borell inequality. If $0<\rho<1$ and

Let $K \subseteq \mathbb{R}^n$ be symmetric and convex, and let $v \in \mathbb{S}^{n-1}$ satisfy $\operatorname{Inf}_v[K] \leq \varepsilon$. A symmetric convex set $L$ can be compa…

Let $n\geq 1$ and let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb R^n$. For $t>0$, let $U_t$ denote the Gaussian diffusion operator, and for a measurable no…