  We investigate Demailly's Conjecture for a general set of sufficiently many
points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing
a lower bound for the Waldschmidt constant of a set of points in projective
spaces. We also study a containment between symbolic and ordinary powers
conjectured by Harbourne and Huneke that in particular implies Demailly's
bound, and prove that a general version of that containment holds for generic
determinantal ideals and defining ideals of star configurations.
