4 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 $B(n)$ 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 $P\subseteq\mathbb R^d$ 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…