Symplectic embeddings II

I’ve been thinking a lot more about symplectic embeddings and the associated combinatorics recently after my first post on the topic, aided by some very useful conversations with my friend Julian and with Hutchings himself! I thought that I would post an extract from my developing TeX document on the subject, which focuses on the role lattice point counts play in the iconic example of ellipsoids. In particular, in viewing a modified version of the ECH capacities associated to an ellipsoid – counting how many capacities stay below each integer value – one gets a direct connection to Ehrhart theory. During another conversation about this today, I realised that one can use the rational version of Ehrhart’s theorem to reduce comparing two sequences of capacities to a finite check, albeit an awkward one best left to computers.

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