639 problems

Let $B_0(r)$ be the ball of radius $r$ centered at the origin, and let $S$ range over simplices containing this ball. Let the Gaussian measure have density proportional to…

Let $C$ be the $\mathbf{\ell}_1$ ball, let $p(w\mid\xi)$ be the posterior density under the uniform prior on $C$, and write $\operatorname{Cov}_{\mathrm{Uni}(C)}(w)$ for the covari…

For every matroid $M$, the Ehrhart $h^*$-polynomial $h^*_{P_M}(t)$ of its matroid base polytope $P_M$ has only real zeros.

For every integer $d\ge 1$ and every integer $r\ge 2$, let $[r]$ denote a set of $r$ vertices, and let $K=[r]*[r]*\cdots*[r]$ be the join of $d+1$ copies of $[r]$. For every contin…

For every integer $n\ge 1$ and every convex body $K\subset\mathbb{R}^n$, let $E$ be a maximum-volume ellipsoid contained in $K$, let $c$ be the center of $E$, and define…

For every integer $n\ge 2$, let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector satisfying $\operatorname{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standar…

Let $\Delta(j,2)$ denote the least integer $d$ such that every collection of $j$ absolutely continuous probability measures on $\mathbb{R}^d$ can be simultaneously equipartitioned…

For every full-dimensional convex body $K\subset\mathbb{R}^n$, $K$ is a simplex if and only if it satisfies the Bézout inequality for mixed volumes. The supplied sources identify t…

There exist an absolutely continuous probability measure $\mu$ on $\mathbb{R}^2$ and a number $t\in(0,1/4)$ such that, for every pair of perpendicular lines in the plane, the four…

Given $n\geq 2$, a coefficient vector $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ that is log-concave, meaning $\alpha_j^2\geq \alpha_{j-1}\alpha_{j+1}$ for…

For $n\ge 2$, let $K\subset\mathbb{R}^n$ be a centrally symmetric convex body, meaning $K=-K$, and let $B\subset\mathbb{R}^n$ be a Euclidean ball satisfying…

Let $X$ and $Y$ be independent identically distributed real-valued random variables with a common log-concave density and finite differential entropy. Define $F:[0,1]\to\mathbb{R}$…

For each $n\ge 2$, let $\mathbb B_n\subseteq\mathbb R^n$ be the Euclidean unit ball and let $K\subseteq\mathbb R^n$ be a convex body with $\mathbb B_n\subseteq K$. If…

For every integer $n\ge 2$, every pair of convex bodies $K,L\subset\mathbb{R}^n$ containing the origin, and every $t\in[0,1]$, prove that…

For every convex body $K\subset\mathbb{R}^n$ and for the admissible higher-order indices, the higher-order mixed-volume inequalities proposed by Schneider should hold. These conjec…

For every number of variables $n$ and every normalized bounded ratio $R$ on Lorentzian polynomials in $n$ variables, the optimal bounding constant of $R$ is at most $2$:…

Let $\mathcal{P}_3$ be the class of convex three-dimensional parallelohedra, namely convex polyhedra that tile $\mathbb{R}^3$ by translations. For every $V>0$ and every…

For every integer $n\ge 2$, every pair of full-dimensional zonoids $K,L\subset\mathbb{R}^n$, and every $u\in S^{n-1}$, one has … where $P_{u^\perp}$ denotes orthogonal projection o…

Let $X$ be a complex normed space and let $n$ satisfy $2\le n<\dim_{\mathbb C}X$. If every pair of $n$-dimensional complex subspaces of $X$ is isometric as metric spaces, then ther…

For integers $d\ge 2$ and $n\ge d+2$, let $F_1,\ldots,F_n\subseteq\mathbb{R}^d$ be sets, each of which is the union of exactly two disjoint, nonempty, closed convex sets. If, for e…

Let $V$ be a normed space over the reals, and let $2\leq k<\dim V$ be an integer. Suppose that all linear $k$-dimensional subspaces of $V$ are isometric to each other. Banach's iso…

Slicing conjecture. The slicing constant is universally bounded:

Let $K$subseteq$\mathbb R^n$ be a symmetric convex body, and define its polar body by … Let $B_2^n=\{x\in\mathbb R^n:\|x\|\leq 1\}$ be the Euclidean unit ball. Ball's conjecture. ……

Kannan–Lovász–Simonovits conjecture. There is a universal constant $C>0$ such that

Let $K$ be a centrally symmetric convex body in $cmathbb{R}^{n}$, and let $\mathcal{P}(K)$ denote its volume product. Let $B_{\infty}^{n}$ be the unit ball of the $\ell_{\infty}$ n…