  In this note we introduce a Waldschmidt decomposition of divisors which might
be viewed as a generalization of Zariski decomposition based on the effectivity
rather than the nefness of divisors. As an immediate application we prove a
recursive formula providing new effective lower bounds on Waldschmidt constants
of very general points in projective spaces. We use these bounds in order to
verify Demailly's conjecture in a number of new cases.
