19 problems

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, and let $K,L$ be origin-symmetric convex bodies in $\mathbb{R}^n$. For every $\lambda\in[0,1]$, Gardner–Zvavitch…

Let $K_1,K_2\subseteq\mathbf{R}^n$ be convex sets whose centroids with respect to the standard Gaussian measure $\mu_n$ coincide. Non-symmetric Gaussian correlation conjecture. Doe…

Let $C=B_2^k(0,w)\times\mathbb{R}^l$, where … Let $K$ be a centrally symmetric convex body with $\gamma(K)=\gamma_{k+l}(C)$ and with inradius $w(K)=w=w(C)$, where $\gamma_{k+l}$ is…

For a convex symmetric set $K\subset\mathbb{R}^n$, define its inradius by … Let $\gamma_n$ denote standard Gaussian measure on $\mathbb{R}^n$. Vershynin's large-width conjecture. F…

B-conjecture. The function $t\mapsto\gamma(e^tK)$ is log-concave. This claim is stated in the source as already proved by Cordero, Fradelizi and Maurey, so it is a theorem rather t…

For $0<p<1$, let $r_p^{(s)}$ denote the greatest $r>0$, if it exists, such that for every dimension $n\geq 2$, every centrally symmetric convex body $K\subset\mathbb{R}^n$, and eve…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$. For a subset $A\subseteq\mathbb{R}^n$, define its $k$-fold Minkowski sum by … Call $A$ balanced if $a\in A$ and…

Let $B_2^n$ denote the Euclidean unit ball in $obreakmathbb{R}^n$, let $gamman$ be standard Gaussian measure, and let $beta(U,V)$ be the vector balancing constant of a pair of con…

A symmetric set $K$ satisfies $K=-K$. Let $K$ and $L$ be symmetric convex bodies in ${\mathbb R}^n$, and let $t\in[0,1]$. Gardner–Zhang's conjecture. … This is a Brunn–Minkowski-ty…

The conjecture. The same right asymptotic order should hold for $r=1$:

Let $\lambda\in(0,1)$, let $K$ and $C$ be origin-symmetric convex bodies in $\mathbb{R}^n$, and let $\gamma_n$ be the canonical Gaussian probability measure. Gardner–Zvavitch conje…

Let $f:\mathbb{R}^n\to\mathbb{R}$ be smooth and satisfy … Assume $\int e^f\,d\gamma_n=1$, let $\mu_f$ be the distribution of $f$ under $\gamma_n$, and let $T_f$ be the monotone rea…

Let $n\geq 1$, let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb{R}^n$, and let $P_s$ be the Ornstein–Uhlenbeck semigroup. For every non-negative measurable f…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, defined by … For $\lambda\in(0,1)$ and sets $A,B\subset\mathbb{R}^n$, write…

Let $\gamma_n$ be the Gaussian probability measure on $\mathbb{R}^n$, and let $K_1$ and $K_2$ be symmetric convex bodies in $\mathbb{R}^n$. The Gaussian Correlation Conjecture. The…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, and let $(Q_t)$ be the Ornstein–Uhlenbeck semigroup defined by … For a non-negative function $f$ of integral $1$,…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, with density $(2\pi)^{-n/2}\exp(-|x|^2/2)$. For closed convex sets $A,B\subset\mathbb{R}^n$ satisfying…

The perturbed-metric conjecture. For every $0<s<1$, there exist constants $C_s,c_s>0$ such that, whenever $\varepsilon(A)<e^{-1/\rho}$,

Let $A$ and $B$ be symmetric convex sets in Euclidean space, and let $\gamma$ denote the standard Gaussian measure. Gaussian correlation conjecture. The source introduces the conje…