1,487 problems

Mahler's conjecture. For every convex body $\Omega\subset\mathbb{R}^N$,

Let $\mu$ be an isotropic log-concave probability measure on $\mathbb{R}^n$, and let $C_P(\mu)$ denote its Poincaré constant, namely the least constant for which the Poincaré inequ…

Let $c$ be a symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. Here $\operatorname{Vol}(X)$ denotes the $2n$-dimensional volume of $X$. Viterbo's conjecture…

Let $m$ be a positive integer and consider all $m$-gons of a fixed area. The affine-orbit part of the Pólya–Szegő polygonal conjecture. The regular $m$-gon minimizes the first Diri…

Kneser–Poulsen conjecture for unequal balls. The simultaneous inequalities

Strict convexity conjecture. The function

Optimal partial plank covering conjecture. There exists a plank $P$ of width $W$ such that, for every such finite family,

Let $c$ be a normalized symplectic capacity and let $X\subset\mathbb{R}^{2n}$ be a convex domain. The Gromov width and cylindrical capacity are extremal among normalized symplectic…

Let $\Omega$ be a convex domain and let $Z_\Omega(s)$ be its tropical zeta function. The image of the map $\Omega\mapsto Z_\Omega$ consists of functions holomorphic on $\Re(s)>1$.…

Let $n\geq 3$ and let $K$ be an $n$-dimensional convex body. A positive homothetic copy of $K$ is a set of the form $x+\lambda K$ with $x\in\mathbb{R}^n$ and $0<\lambda<1$. Hadwige…

Petty's projection conjecture. The quantity $\mathcal{F}(K)$ is minimized over all convex bodies $K$ in $\mathbb{R}^n$ if and only if $K$ is an ellipsoid.

A plank $P$ is a region between two parallel hyperplanes in $\mathbb{R}^d$, and its relative width with respect to a convex body $K$ is … where $u$ is the normal of $P$. Let…

Let $M_t=X(\mathbb{S}^n,t)\subset\mathbb{R}^{n+1}$ be a family of smooth, strictly convex, closed hypersurfaces evolving by Gauss curvature flow, with $K$ the Gauss curvature and…

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, and let $K,L$ be origin-symmetric convex bodies in $\mathbb{R}^n$. For every $\lambda\in[0,1]$, Gardner–Zvavitch…

Let $K$ be a convex body in $\mathbb{E}^3$. The density of a packing is the proportion of space occupied by the translates of $K$, and the densest packing density of $K$ is the sup…

Betke-Henk-Wills conjecture. For any $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$,

For $1\leqslant k\leqslant n-1$, let $G_{n,k}$ be the Grassmannian of $k$-dimensional subspaces of $\mathbb{R}^n$, let $P_E$ denote orthogonal projection onto $E$, and let…

Let $K\subseteq\mathbb{R}^d$ be a convex body centred at the origin, meaning that its centre of mass is the origin. Assume that the interior of $K$ contains no lattice point other…

Let $K,L\subseteq\mathbb{R}^n$ be symmetric convex sets, and let their geometric mean be the set obtained by combining their support functions geometrically. Denote $n$-dimensional…

Let $K=-K\batchrel{\subset}\mathbb{R}^n$ be an origin-symmetric convex body containing the origin in its interior, and let $K^\circ$ be its polar body. Write…

Central limit conjecture. There exists a sequence of positive numbers $(\sigma_n^2)_{n\ge1}$ such that

Slicing conjecture. The slicing constant is universally bounded:

Equilateral triangle torsional-rigidity conjecture. The equilateral triangle of unit width minimizes the torsional rigidity among shapes having unit minimal width.

Let $K\subset\mathbb{R}^n$ be a convex body, and let $V$ denote mixed volume, with $K[k]$ denoting $k$ copies of $K$. Godbersen's conjecture. For each convex body…

Let $n\ge1$ and $k\ge2$. Let $\rho:\mathbb R\to\mathbb R_+$ be non-increasing, and let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R_+$ be even integrable functions. For … assume … for a…