12 problems

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 $X_1,\dots,X_n$ be independent random vectors with values in a normed space $\mathcal{S}$. Let $\tilde{\mathcal{Q}}(X,1)$ denote the supremum of the probability that $X$ lies i…

Let $X_1,\dots,X_n$ be independent random vectors in a normed space, and let $\tilde{\mathcal{Q}}(X,1)$ denote the supremum of the probability that $X$ lies in a set of diameter at…

Let $\bm{X}=(X_1,\dots,X_k)$ be a random vector with CDF … for all $\bm{x}=(x_1,\dots,x_k)\in\mathbb{R}^k$, and let $\epsilon$ be an arbitrary positive number. The paper proposes e…

Gaussian minimum conjecture. One has

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{R}^d$ satisfying … A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and…

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{Z}^d$. A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and all…

Let $X$ denote the random vector considered in the paper, and let the two parts of Question 1 ask whether the relevant point-separation and stable point-separation properties hold…

Let $\mathbf{y}\in\mathbb{R}^n$ be an isotropic random vector with a log-concave density. Thin shell conjecture. There is a universal constant $c>0$ such that, for $t>0$, … This is…

Let $X$ be an isotropic log-concave random vector and let $E$ be a corresponding exponential random vector. For a norm $\|\cdot\|$ on the ambient Euclidean space, compare the expec…

Uniform quadratic sampling conjecture. With high probability or in expectation,

Let $X$ be a normalized uniformly distributed random vector. For mutually unbiased bases, let $m_1,m_2,\ldots,m_k$ denote the relevant maximum second-moment quantities, and let…