The polyhedral Tammes problem (Q5943214)

The Tammes problem asks for the maximum of the shortest spherical distance among \(n\) points on the unit sphere. The authors consider a more general version of this problem, the polyhedral Tammes problem: For given integers \(n\geq d+1\geq 3\), find the maximum of the shortest distance between any two of the \(n\) vertices of a convex \(d\)-polytope with diameter 1 in \(E^d\). This is equivalent to the problem: Find the minimum diameter \(F(n,d)\) of convex \(d\)-polytopes with \(n\) vertices having pairwise distances \(\geq 1\) in \(E^d\). Another similar problem is: Find the minimum diameter \(f(n,d)\) of convex \(d\)-polytopes with \(n\) vertices having edge lengths \(\geq 1\) in \(E^d\). Obviously \(1\leq f(n,d)\leq F(n,d)\). The authors prove among others the following results: \(\bullet\) \(f(n,d)\leq 3\) for all \(n\geq d+1\geq 3\). \(\bullet\) \(f(6,3)=\sqrt 2\).