On a conjecture of Demailly and new bounds on Waldschmidt constants in PN

In the present note we prove a conjecture of Demailly for finite sets of sufficiently many very general points in projective spaces. This gives a lower bound on Waldschmidt constants of such sets. Waldschmidt constants are asymptotic invariants of subschemes receiving recently considerable attention [2], [1], [6], [11], [14].