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 centralized generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic $S^1$-index, we propose a definition of generalized Zoll convex bodies and prove that this definition is equivalent to the usual one in the smooth setting. Moreover, we show how 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 such a convex body must be generalized Zoll, and in particular Zoll if its boundary is smooth. Finally, we discuss some examples.