On the Density Function on Moduli Spaces of Toric 4-manifolds

The optimal density function assigns to each symplectic toric manifold M a number 0 < d ≤ 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.