54 problems

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

Slicing conjecture. The slicing constant is universally bounded:

Let $\Lambda\subset\mathbb{R}^d$ be a $d$-dimensional lattice. For a convex body $K\subset\mathbb{R}^d$, define its lattice-normalized volume by … Consider the maximum of…

Let $K\subseteq \mathbb R^n$ be an origin-symmetric convex body. An origin-symmetric convex body $L\subseteq \mathsf{A}K$ is an $O(1)$-perturbation of $\mathsf{A}K$ when its volume…

Let $\phi$ be a smooth, strictly convex norm on $\mathbf{R}^{n+1}$, with Wulff shape $\mathcal{W}^\phi=\{x\in\mathbf{R}^{n+1}:\phi^*(x)=1\}$. Let $X_0:M\to\mathbf{R}^{n+1}$ be a cl…

Let $K$ be an $n$-dimensional convex body in $cmathbb{R}^n$. The translative constant volume property means that the volume of the convex hull of any touching pair of translates of…

Lattice-packing conjecture. The densest packings of the centrally symmetric Platonic and Archimedean solids are given by their corresponding optimal lattice packings.

Let $E$ be a non-symmetric convex body in ${\bf R}^d$, let $\lambda(x)$ denote the density of its equilibrium measure in the interior, and let $\widetilde{G}(E,x)$ be the convex hu…

Let $K$ be a convex body in a normed space, let $x\in\operatorname{int}K$, and let $B_n(K,x)$ denote the Bernstein factor at $x$ for polynomials of degree at most $n$. Write $w(K)$…

Let $K\subset\mathbb{R}^n$ be a convex set of volume one, and let $X$ be uniformly distributed inside $K$. Anttila–Ball–Perissinaki conjecture. For some non-zero vector…

Let $\Om\subset\mathbb{R}^3$ be a strictly convex body, and let $\Sg$ be an isoperimetric hypersurface inside $\Om$. Ros's disk conjecture. The hypersurface $\Sg$ should be homeomo…

Relative discrepancy conjecture. In dimensions $d\geq 5$, the relative error satisfies $\Delta_{\Omega}(t)=O(t^{-2})$ as $t\to\infty$.

Let $K\subset \mathbb{E}^3$ be a convex body and let $N\in K$. For a plane $\Gamma$ with $\Gamma\cap\operatorname{int}K\not=\emptyset$ and $N\notin\Gamma$, write…

Grünbaum distance equality characterization conjecture. Equality $d_G(K,L)=n$ occurs only when one of $K$ and $L$ is a simplex.

Fix $d\in\mathbb{N}$, let $\mathcal{K}^d$ be the family of convex compact subsets of $\mathbb{R}^d$, and let $\mathcal{K}_j^d=\{K\in\mathcal{K}^d:\,\dim(K)\geq j\}$ for…

Conjectural equivalence. The following conditions are equivalent:

$L_p$-Brunn–Minkowski conjecture. Does it necessarily follow that

Lattice covering density conjecture. For all $K\in\mathcal{K}^{n}$, there is an absolute constant $c>0$ such that

Let $M$ be a symmetric convex body, and let $x\notin M$. Say that $M$ looks symmetric from $x$ when its graze from $x$ has the corresponding symmetry property. Bianchi–Gruber conje…

Let $M,K$ be convex bodies in $\mathbb{R}^n$ with $M\subset\operatorname{int}K$ and $n\geq 3$. For $x\in\operatorname{bd}K$, let the graze $\Sigma(M,x)$ be the set of contact point…

Ball density–kissing-number conjecture. One may conjecture that

Cross-polytope kissing number conjecture. In $\mathbb{E}^4$,

Cross-polytope lattice density conjecture. In $\mathbb{E}^4$,

Let $B^n$ denote the unit ball in Euclidean space $\mathbb{E}^n$, and let $\kappa^*_{\alpha}(B^n)$ be the generalized lattice kissing number at parameter $\alpha$. Conjecture 8.1.…

Let $B^n$ denote the unit ball in Euclidean space $\mathbb{E}^n$, and let $\kappa^*_{\alpha}(B^n)$ be the generalized lattice kissing number at parameter $\alpha$. For the Leech la…