9 problems

Boundary-area conjecture. There are absolute constants $c_1,c_2>0$ such that

Prochno–Schütt–Werner conjecture. The same lower bound should hold without the assumption $N\geq (c_1d)^{\frac{d-1}{2}}$.

Bárány's vertex-removal conjecture. For every $n\in\mathbb N$ there is a constant $c_n\in(0,\infty)$ such that, for every such polytope $P$ with $N$ vertices, some vertex $x$ satis…

Generalized-curvature extension conjecture. It is generally conjectured that Theorem Gruber's best-approximation result holds for arbitrary convex bodies when the classical Gauß-Kr…

Let $\operatorname{ldel}_{n-1}$ denote the Laguerre–Delone tiling number in $\mathbb{R}^{n-1}$. For $n=3$, Gruber's conjecture. … These constants arise from Laguerre–Delone tilings…

Fix $n\in\mathbb{N}$ with $n\geq 2$ and $N\geq n+1$, and let $K$ be a convex body in $\mathbb{R}^n$. For a polytope $P$ with at most $N$ facets, write $\Delta_v(P,K)$ for the symme…

Let $D_n$ be the Euclidean unit ball in $\mathbb{R}^n$, let $N$ be a positive integer, and let $P^b_{n,N}$ be a polytope with at most $N$ facets that is best-approximating for…

Let $D_n$ be the Euclidean unit ball in $\mathbb{R}^n$, let $A\geq 10$ be fixed, and let $P^b_{n,A^n}$ be a polytope with at most $A^n$ facets that is best-approximating for $D_n$…

Regular-pentagon conjecture. One has