*Accepted Paper*

**Inserted:** 23 jun 2015

**Last Updated:** 23 jun 2015

**Journal:** Adv. Geom.

**Year:** 2015

**Abstract:**

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 then 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 $4$.

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