Remarks on the systoles of symmetric convex hypersurfaces and symplectic capacities

TL;DR

This paper investigates the relationship between systoles and symmetric systoles of convex hypersurfaces in symplectic space, establishing bounds using Floer theory and analyzing explicit examples.

Contribution

It introduces a uniform upper bound on the ratio of systole to symmetric systole for convex hypersurfaces invariant under anti-symplectic involution, using symplectic capacities.

Findings

Established a uniform upper bound for the systole ratio.

Provided explicit examples illustrating the ratio.

Applied Floer theory to analyze symplectic capacities.

Abstract

In this note we study the systoles of convex hypersurfaces in $\mathbb{R}^{2n}$ invariant under an anti-symplectic involution. We investigate a uniform upper bound of the ratio between the systole and the symmetric systole of the hypersurfaces using symplectic capacities from Floer theory. We discuss various concrete examples in which the ratio can be understood explicitly.