Let

be a set of

general points in the projective space

over an algebraically closed field

. For each positive integer

, let
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denote the smallest degree of nonzero homogeneous polynomials in

that vanish to order at least

at every point of

. The Waldschmidt constant
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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