On some metric and combinatorial geometric problems

AbstractLet x1,…,xn be n distinct points in the plane. Denote by D(x1,…,xn) the minimum number of distinct distances determined by x1,…,xn. Put ƒ(n) = min D(x1,…, xn). An old and probably very difficult conjecture of mine states that ƒ(n) >cn(log n)12. ƒ(5) = 2 and the only way we can get ƒ(5) = 2 is if the points form a regular pentagon. Are there other values of n for which there is a unique configuration of points for which the minimal value of ƒ(n) is assumed? Is it true that the set of points which implements ƒ(n) has lattice structure? Many related questions are discussed