Linear bounds for lengths of geodesic loops on Riemannian 2-spheres

Abstract

Let $M$ be a closed surface diffeomorphic to $S^2$ endowed with a Riemannian metric. Denote the diameter of $M$ by $d$. We prove that for every $x \in M$ and every positive integer $k$ there exist $k$ distinct geodesic loops based at $x$ of length $\le 20kd$.