Small convex polytopes with long edges and many vertices (Q1423582)

Let \(\delta^d_n\) be the infimum of the (Euclidean) diameters of all convex polytopes in~\(\mathbb{R}^d\) with~\(n\) vertices which do not have any edge of (Euclidean) length less than one. Let \(\overline{\delta}^d\) and \(\underline{\delta}^d\) be the supremum limit and the infimum limit of \(\delta^d_n\), respectively, as \(n\rightarrow\infty\). It is easy to see that \(\overline{\delta}^2=\underline{\delta}^2=\infty\). The present paper shows however, that for \(d\geq 3\) the limits are finite. In fact, it is proved that \(\overline{\delta}^3=\underline{\delta}^3=3\), \(\overline{\delta}^d=\underline{\delta}^d=\sqrt{3}\) for \(d\geq 7\), and \(\overline{\delta}^d\leq 2\), \(\underline{\delta}^d\geq\sqrt{3}\) for \(4\leq d\leq 6\) hold. The upper bounds are derived by a skillful lifting procedure that produces from a \(d\)-polytope~\(K\) (containing two distinguished points that have distance at least one from each other as well as from all vertices of~\(K\)) some appropriate \((d+2)\)-polytope whose diameter is not larger than that of~\(K\). The lower bounds are based on arguments showing that polytopes with many vertices but without edges shorter than one must have large angles in their \(1\)-skeleta.