8 problems

Let $S \subset \mathbb{Z}^d$ be a bounded set, and let $\beta_{\mathbb{Z}}(S)$ denote its discrete Borsuk partition number. Let $\operatorname{conv}(S)$ be the convex hull of $S$,…

Let $S$ be a bounded set in a $d$-dimensional normed space. Boltyansky–Gohberg conjecture. The set $S$ can be partitioned into at most $2^d$ subsets, each having strictly smaller d…

For real $p\geq 1$, let $\ell_p^d$ be $\mathbb R^d$ equipped with the norm … and let $\ell_\infty^d$ use the norm $\|\mathbf x\|_\infty=\max_i|x_i|$. Write $b(\ell_p^d)$ for the co…

Let $\mathbb M^d$ be a $d$-dimensional Minkowski normed space, and let $b(\mathbb M^d)$ be the least positive integer $k$ such that every bounded non-singleton subset of…

Let $B(n)$ be the maximum, over all norms on $\mathbb{R}^n$, of the Borsuk number of $(\mathbb{R}^n,\left\Vert\cdot\right\Vert)$, where the Borsuk number is the smallest number of…

Let $u(d)$ be the least $t$ such that the unit ball in $\mathbb R^d$ can be covered by $d+1$ sets of diameter at most $t$. Strong Borsuk covering conjecture. Every set of diameter…

Let $P\subseteq\mathbb R^d$ be a polytope with $m$ facets. A covering by smaller-diameter sets is a cover of $P$ by sets whose diameters are strictly smaller than the diameter of…

Let $K\subseteq\mathbb R^d$ be a set of points, let $D$ be the set of pairs of points in $K$ at the diameter distance, and call $(K,D)$ general if every continuous deformation of t…