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