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Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. …

Let $\Delta_n := \{x \in \mathbb{R}^n | x \ge 0, \sum_{1 \le i \le n}x_i = 1\}$ be the $n$-simplex. For $a, b \in \mathbb R^n$, with $\Delta_n \not \ni a$, consider the problem of computing the follo …

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that $\ …

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set $$ \alpha := \sup_{(x,y) \in C} ax + b y. $$ Question. In …

Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT... Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's sa …

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties 1. Symmetry: $\zeta \overset{d}{=} - \zeta$. 2. Small-ball probability: there exists …

Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the …

Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the fun …

Let $C$ be a (nonempty) convex compact subset of $\mathbb R^n$. General question: Under what conditions on $C$ does the support function $$\sigma_C(x) := \sup_{y \in C}x^Ty $$ define a norm on $\mat …

As far as I know, the most popular way to sample from a polytope (in H-representation) \begin{equation} \mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\} \end{equation} i …

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ever …

Let $n$ be a positive integer, and suppose $f$ is a probability distribution on the $2^n$ subsets of $[\![n]\!] := \{1,\ldots,n\}$. What is a "natural" way to extend $f$ to a distribution $\bar{f}$ on …

Given an $m \times n$ matrix $A$ and a vector $c \in \mathbb R^n$, define $\eta(A,c) \ge 0$, $$ \eta(A,c) := \sup_{u \in \mathbb R^n} u^\top c - \frac{1}{2}\|u\|^2 - \|Au\|. $$ Note that $\eta(A,c) = …

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$. Question. What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution o …

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = …