# minimal diameter of sphere with sectional curvature bounded above

Assuming that $g$ is a metric on the n-dimensional sphere $S^n(n\geq 3)$ satisfying sectional curvature bounded: $|K(g)|\leq 1$, is the diameter of $g$ at least $\pi$? How about if we only assume that $K(g)\leq 1$?

• I don't have time for a full answer now, but there is a paper by Buser and Gromoll (with title something about almost non-negative metrics on $S^3$) which shows that under the assumption $K(g)\leq 1$, the diameter may be as small as you wish. Mar 2 '17 at 19:52

If $0<h\leq K \leq H$ everywhere, then it is a result of Mei-Chin Ku (PAMS, 1976) that
$$D\geq \frac{2\pi}{3\sqrt{H}}.$$