21 problems

Let $q>0$ be real, let $a_1,\dots,a_n$ be non-negative real numbers satisfying … and let $X_1,\dots,X_n$ be independent identically distributed standard exponential random variable…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $\mu_K$ be the equilibri…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $g$ be the Green's funct…

Let $p$ and $q$ be real-rooted polynomials of degree $d$, and let $\f$ be in $C^{4}(\mathbb{R})$ with $f^{(4)}\geq 0$. Write $\mu_p$ and $\mu_q$ for the probability measures assign…

Let $\mathscr{E}_1,\mathscr{E}_2,\dots$ be independent exponential random variables, let $G\sim N(0,1)$, and let $p_0$ be the crossover parameter used in the paper. Consider all fi…

Hunter's conjecture. The global minimum of $h_{2k}$ on this normalized sphere is attained at $\tilde{a}$.

Endpoint maximizer conjecture. The supremum above is attained at $x=1$.

Let $\Gamma$ have density $e^{-(x+1)}\mathbf{1}_{[-1,\infty)}$, let $\Gamma'$ be an independent copy of $\Gamma$, and for $0\leq s\leq 1$ define … For $p\geq q\geq 1$, consider the…

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

Extremal moment-ratio conjecture. One has

Let $U_1,U_2,\ldots$ be independent random variables uniformly distributed on $[-1,1]$, let $p\in(0,1)$), and let $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$. The sharp $L_p$-$L_2$ moment…

Lewicki–Skrzypek conjecture. For every $n\in\mathbb{N}$,

Gaussian chaos moment upper-bound conjecture. Under the assumptions of the stated moment estimate, for every $p\geq1$,

Let $C_{p,q,d}$ denote the smallest constant such that, for every $d$-homogeneous polynomial $h:\{-1,1\}^{n}\to\mathbb{C}$, one has … For chaoses with coefficients in arbitrary nor…

The $q$-moment conjecture. If $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, then

Subgaussian multiplier estimate under finite-range moment growth. Under this moment assumption, a subgaussian-type estimate like

Non-Lipschitz moment conjecture. There is a constant $C_D$ such that

Let $d\geq 2$, let $(G_1,\ldots,G_d)$ be a $d$-dimensional real-valued centered Gaussian vector, and let $m\geq 1$ be an integer. Gaussian Product Conjecture. One has … This is an…

The conjecture. The moment inequality

Comparable weak and strong moments conjecture. Every symmetric log-concave probability measure on $\mathbb R^n$ satisfies $\mathrm{CWSM}(C)$.