A Zoll counterexample to a geodesic length conjecture

TL;DR

This paper constructs a counterexample on the 2-sphere showing that the previously conjectured inequality relating the length of the shortest closed geodesic and the diameter does not always hold, challenging existing assumptions.

Contribution

It provides the first explicit Zoll counterexample to the L<2D inequality, demonstrating the non-optimality of the round metric for this ratio.

Findings

Counterexample disproves the L<2D conjecture

Round metric is not optimal for the L/D ratio

Uses Guillemin's theorem on Zoll surfaces

Abstract

We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D.