90 problems

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

Atiyah–Sutcliffe Conjecture 2. One has

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

Atiyah's conjecture. For every $\mathbf{x}\in C_n(\mathbb{R}^3)$, the polynomials $p_1,\ldots,p_n$ are linearly independent over $\mathbb{C}$. This is the original Atiyah problem o…

Equality conjecture. There is equality if and only if $K$ is a ball.

Let $K\subset\mathbb R^n$ be a convex body, and for $m\geq 1$ define … Set … In the range $n\geq3$ and $m\geq2$, Schneider's conjecture. … with equality if and only if $K$ is an el…

Let $K\subset\mathbb R^n$ be a convex body, and let $D^m(K)\subset\mathbb R^{nm}$ denote its $m$th-order difference body. For fixed volume $|K|_n$, Schneider conjectured that the v…

Let $K$ and $C$ be the convex bodies appearing in the affine-ratio bounds, and let $d_{GR}(K,C)$ denote their geometric representation distance. Equality in the upper bound … requi…

Böröczky–Lutwak–Yang–Zhang's Log-Brunn–Minkowski conjecture. The inequality

Let $n>k\geq 1$, let $\Box^n=[-1,1]^n$, and let $H$ be a $k$-dimensional linear subspace. Write $C_{\Box}(n,k)2^k$ for the maximum volume among sections $\Box^n\cap H$ that are aff…

Let a convex closed curve have perimeter $L$, area $A$, and largest diameter $D$. Consider the equation … The source refers to a system of equations intended to provide an inverse…

Let $1\leq \ell\leq n/2$, let $i=\ell+1$ or $i=\ell+2$, and let $K_\ell$-symmetric body $A$ and o.s. star body $B$ be given. The notation $G_{n,i}$ denotes the Grassmannian of $i$-…

Isoembolic ball and sphere volume rigidity conjecture. (a) For every $r\leq \operatorname{inj}(X)/2$,

Let $m$ points lie on a line and let the remaining $n=N-m$ points form a regular $n$-gon in a plane perpendicular to the line, with centroid on the line. Let…

Let $\Pi_n$ be an $n$-gon with side lengths $a_1,\ldots,a_n$, perimeter $L_n$, and area $A_n$. Define … and set $\varphi_n=A_n/P_n$. Levy's ratio bounds conjecture. For any $n$-gon…

Mean-chord inequalities conjecture. Without regularity restrictions, $C_L^{\pm p}(u)$ holds for any $p\leq2$, and the same is true for $D_{N,\ell}^{\pm p}(m)$. Moreover, the inequa…

Let $\Pi_n$ be an $n$-gon with side lengths $a_1,\ldots,a_n$, perimeter $L_n$, and area $A_n$. Define its pseudo-area by … For the regular $n$-gon, let … Define the ratio…

Simplex extremality conjecture. An $n$-dimensional regular simplex is an extreme body for the optimal upper bound of the ratio $R_{n-i+1}(K)/r_i(K)$ among convex bodies $K$.

Cordero-Erausquin's conjecture. The inequality

Let $K\text{ be a convex body in }\mathbb{R}^n$ and let $(\sigma_1,\dots,\sigma_m)$ be an $s$-cover of $\sigma\subset[n]$. For each $\tau\subset[n]$, let $H_\tau$ denote the coordi…

Mask conjecture. A minimizer is a "mask": its boundary is composed of arcs of circles with three different curvatures.

The volume inequality. One has

Even log-concave dimensional inequality conjecture. An improvement of

Let $T$ be a tree, meaning a connected circuit-free non-oriented finite simple graph, and let $C_T(\mathbb{R}^3)$ be its configuration space. Let $\mathcal{A}_T$ be the associated…