The length of the shortest closed geodesic on positively curved 2-spheres

Besse projective spaces with many diameters

“…We note that the bound L(S n , g) ≤ 2 √ δ D(S n , g) is also achieved in [2] via different techniques, albeit in the more restrictive setting of pinching constant 0.83…”

“…Nabutovsky and Rotman [14] and independently Sabourau [16] developed new techniques to prove that L(S 2 , g) ≤ 4D(S 2 , g). By imposing bounds on curvature, the authors have previously shown in [2] that L(S 2 , g) ≤ 3D(S 2 , g) for non-negative metrics on the 2-sphere and that L(S 2 , g) ≤ 2 √ δ D(S 2 , g) for δ > .83 pinched metrics on the 2-sphere.…”