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Let $M$ be a Riemannian manifold with $\text{Sec}\ge 0$. From Topogonov Theorem follows that for every $p \in M$ the quantity $$ \frac{\text{Diam}(B_r(p))}{r} $$ is non-increasing in $(0,\infty)$. Does this hold also if we only assume $\text{Ric}\ge 0$? I suspect that this is no longer true, however I am not able to build a counterexaple.

Clarification: $\text{Diam}(B_r(p)):=\sup_{x,y\in B_r(p)}d(x,y).$

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    $\begingroup$ It is a theorem due to Abresch-Gromoll that diameter growth is linear. Any complete Riemannian manifold with $Ric \geq 0$ has diameter growth of order $O(r)$ with respect to any point. $\endgroup$ – C.F.G Sep 11 at 13:06
  • $\begingroup$ The notion of diameter growth in that paper is not the one I am interested in, here by $\text{Diam}(B_r)$ I mean the supremum of $d(x,y)$ among all $x,y \in B_r$, which clearly implies $\text{Diam}(B_r)\le 2r.$ $\endgroup$ – mrprottolo Sep 11 at 13:19

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