A Bound on the Symplectic Systolic Ratio of Polytopes in Even-Dimensional Euclidean Space

Abstract: Symplectic capacities are invariants in symplectic geometry that are used to obstruct symplectic embeddings. From a certain symplectic capacity, the Ekeland-Hofer-Zehnder capacity, one can construct the systolic ratio, which measures the difference in the capacity and the Euclidean volume. The systolic ratio of a unit ball is always 1. It was conjectured by Viterbo that the systolic ratio is bounded by 1. This conjecture was disproven by Haim-Kislev recently, and now it is open to determine what bounds one may obtain on the systolic ratio if not 1. In this paper, the author investigates bounds on the systolic ratio of polytopes. In particular, a sharp bound on the systolic ratio of simplices in any dimension is obtained, with the dependence on dimension made explicit.