7 problems

Let $\Diamond^n=\operatorname{conv}\{\pm e_1,\dots,\pm e_n\}\subset\mathbb R^n$ be the standard cross-polytope, let $E$ be a $k$-dimensional subspace, and let $P_E$ denote orthogon…

Covering product conjecture. One has

Let ${\bf p}=(p_1,\ldots,p_N)$ and ${\bf q}=(q_1,\ldots,q_N)$ be configurations of $N$ points in ${\mathbb E}^d$. The configuration ${\bf q}$ is a contraction of ${\bf p}$ if…

Let finitely many balls in Euclidean space have centers, and suppose the centers undergo a contraction, meaning that every pairwise distance between centers does not increase. Grom…

Let $M$ be a closed Riemannian $n$-manifold, let $\inj(M)$ denote its injectivity radius, let $\alpha_n$ be the volume of the canonical $n$-sphere, and let $B(r)$ be a ball of radi…

Let $S=\{v_1,\ldots,v_n\}$ be an $(n,k)$-frame spanning $\mathbb{R}^k$, let … and define $P=\operatorname{co}\{\pm v_1,\ldots,\pm v_n\}$ and…

Let $H_k$ be a $k$-dimensional subspace and let $\Diamond^n\mid H_k$ denote the corresponding affine section of the $n$-dimensional cross-polytope, while $\Diamond^k$ is the $k$-di…