Generalized convex toric domains and symplectic embedding problems

A convex toric domain  is a 4-dimensional subset of , defined as the preimage of a bounded convex region  in the positive quadrant of  under the moment map. We consider how geometric features of  such as the curviness of its boundary and its affine perimeter impact symplectic packing problems. Some of our results come from considering the asymptotics of the ECH capacities. These capacities are known to obey a Weyl law and thus detect the volume of . We show that their subleading asymptotics detect the affine perimeter of . We'll discuss how these asymptotic results lead to new applications in symplectic embedding problems. This is based on joint work with Dan Cristofaro-Gardiner and Dusa McDuff.