Systolic $S^1$-index and characterization of non-smooth Zoll convex bodies

TL;DR

This paper introduces the systolic $S^1$-index as a symplectic invariant for convex bodies, characterizes generalized Zoll convex bodies, and explores their properties and examples within symplectic geometry.

Contribution

It defines the systolic $S^1$-index, introduces generalized Zoll convex bodies, and links these to Gutt-Hutchings capacities, providing new insights into symplectic invariants of convex bodies.

Findings

The systolic $S^1$-index is a symplectic invariant.

Generalized Zoll convex bodies are characterized by Gutt-Hutchings capacities.

The space of generalized Zoll convex bodies is closed in the space of all convex bodies.

Abstract

We define the systolic $S^1$-index of a convex body as the Fadell-Rabinowitz index of the space of generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic $S^1$-index, we introduce the notion of generalized Zoll convex bodies and prove that this definition coincides with the classical one when the convex body satisfies the uniqueness of systoles property, that is, when through every point passes at most one systole. Moreover, we show that generalized Zoll convex bodies can be characterized in terms of their Gutt-Hutchings capacities, and we prove that the space of generalized Zoll convex bodies is closed in the space of all convex bodies. As a corollary, we establish that if the interior of a convex body is symplectomorphic to the interior of a ball, then the convex body is generalized Zoll, and in particular Zoll if it satisfies the uniqueness of systoles property. Finally, we discuss several examples.