31 problems

Let $(X_1,\dots,X_n)$ be a centered Gaussian random vector, and let $\alpha_j$ be non-negative real numbers for $j=1,\ldots,n$. Gaussian product inequality conjecture. … This is a…

Let $(X_1,\ldots,X_d)$ be a $d$-dimensional real-valued centered Gaussian random vector, and let $\alpha_j$, $j=1,2,\ldots,d$, be nonnegative real numbers. Li–Wei's improved Gaussi…

Let $cnu$ be a rotationally invariant measure on $cmathbb{R}^{n}$, absolutely continuous with respect to Lebesgue measure, with density $f(|x|)$ for a nondecreasing function…

Let $c(F)$ be a separable Banach space, let $cmu$ be a centered Gaussian measure on $F$, and let $A,B$ be Borel subsets of $F$. Ehrhard's conjecture. Inequality (Ehrhard's inequali…

Let $m=d+1$ equiprobable signals of equal energy be transmitted as points in $\mathbb R^d$ through an additive white Gaussian noise channel, and let the receiver use maximum-likeli…

Let $\Omega_0$ and $\Omega_1$ be open, bounded, centrally symmetric subsets of $\mathbb{R}^n$, and let $t\in[0,1]$. Write … The Gaussian torsional rigidity of a set $\Omega$ is den…

Let $\Omega_0$ and $\Omega_1$ be convex sets containing the origin, and let $\Omega_t=(1-t)\Omega_0+t\Omega_1$ for $t\in[0,1]$. The Gaussian measure is denoted by $\gamma$. Garcia-…

Let $b3$ denote standard Gaussian measure on $a5^n$, and let $K,L\to a5^n$ be origin-symmetric convex sets. The Minkowski sum is $K+L={\,x+y:x\in K, y\in L}$, and $K\cap L$ is thei…

Let $\boldsymbol{X}=(X_1,\ldots,X_d)$ be a real centered Gaussian random vector, where $d\in\mathbb{N}$, and let $m\in\mathbb{N}$. Gaussian product inequality conjecture. … This co…

Let $d>0$ and let $K,T\subseteq \mathbb R^d$ be origin-symmetric convex sets, and let $\gamma_d$ denote the usual Gaussian probability measure. Teheranchi's strong Gaussian correla…

Let $X_1,\dots, X_n$ be zero-mean jointly Gaussian real random variables, and let $s_1,\dots, s_n, t_1,\dots, t_n \in [0,\infty)\cup \{\infty\}$. Teheranchi's strong Gaussian corre…

Let $K$ be a symmetric convex body in $\mathbb R^n$. Its maximal intersection position is the condition that, for every volume-preserving linear operator $T$ on $\mathbb R^n$, one…

Let $n\in\mathbb{N}$ and let $p_1,\dots,p_n\in 2\mathbb{N}$. For a real centered Gaussian vector $(X_1,\dots,X_n)$, the statement $\mathbf{GPI}_n(p_1,\dots,p_n)$ means … with equal…

Let $\boldsymbol{X}=(X_1,X_2,\dots,X_d)$ be a centered Gaussian random vector, and let $m\in\mathbb N$. Gaussian product inequality conjecture. For every $m\in\mathbb N$, one has ……

Let $m_1,m_2,m_3\in\mathbb{N}$, and let $(X_1,X_2,X_3)$ be a centered Gaussian random vector. Three-dimensional Gaussian product inequality conjecture. … The equality sign is asser…

Let $n$ be a positive integer, let $y_1,ots,y_n$ be non-negative real numbers, and let $(X_1,ots,X_n)$ be an $n$-dimensional real-valued centered Gaussian random vector. Gaussian p…

Let $W$ be distributed as $\operatorname{Wishart}_p(2\alpha,\Sigma/2)$, where $2\alpha\in\mathbb{N}\cup(p-1,\infty)$, $\Sigma$ is a $p\times p$ symmetric positive definite matrix,…

Let $\boldsymbol{Z}=(Z_1,\ldots,Z_d)$ be a centered Gaussian random vector with $d\geq 2$. For reals $\alpha_1,\ldots,\alpha_d\in[0,\infty)$ and an integer $d_1\in\{1,\ldots,d-1\}$…

Let $n\ge 3$, let $m_1,\dots,m_n\in\mathbb{N}$, and let $U_1,\dots,U_n$ be independent standard Gaussian random variables. Define the polynomial $H$ on $\mathbb{R}^{n(n-1)/2}$ by ……

Let $n\ge 3$, let $m_1,\dots,m_n\in\mathbb{N}$, and let $(X_1,\dots,X_n)$ be a centered Gaussian random vector. The even Gaussian product inequality conjecture states that … Moreov…

Combinatorial positivity conjecture.

Three-dimensional Gaussian product inequality conjecture.

Under Assumption … sigma^Y{i,j}geq sigma^X{i,j}geq 0 … operatorname{Cov}(log SN({mathbf G}),pi({mathbf G})pj({mathbf G}))leq 0. … ; its resolution is not indicated in the supplied…

Let $n$ be a positive integer, let $\gamma$ denote the standard Gaussian measure on $\mathbb R^n$, and let $K,L\subseteq\mathbb R^n$ be symmetric convex sets. For $p\geq 0$, define…

Gaussian minimum conjecture. One has