  The optimal density function assigns to each symplectic toric manifold $M$ a
number $0 < d \leq 1$ obtained by considering the ratio between the maximum
volume of $M$ which can be filled by symplectically embedded disjoint balls and
the total symplectic volume of $M$. In the toric version of this problem, $M$
is toric and the balls need to be embedded respecting the toric action on $M$.
The goal of this note is first to give a brief survey of the notion of toric
symplectic manifold and the recent constructions of moduli space structure on
them, and recall how to define a natural density function on this moduli space.
Then we review previous works which explain how the study of the density
function can be reduced to a problem in convex geometry, and use this
correspondence to to give a simple description of the regions of continuity of
the maximal density function when the dimension is $4$.
