Let
be a set of
general points in the projective space
over an algebraically closed field
. For each positive integer
, let
%24&chf=bg,s,333333&chco=ffffff)
denote the smallest degree of nonzero homogeneous polynomials in
that vanish to order at least
at every point of
. The Waldschmidt constant
%24&chf=bg,s,333333&chco=ffffff)
of
is defined by the limit
Demailly conjectured that
Malara, Szemberg, and Szpond proved that Demailly's conjecture holds if
In a joint work with Yu-Lin Chang, we improve their result and show that Demailly's conjecture holds if
where
is the fractional part of
. In particular, if
, namely
, then Demailly's conjecture holds for all
.
We also show that the inequality conjectured by Demailly can be derived from another conjectural inequality
which was conjectured by Iarrobino (and by Nagata for
) to hold if
