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Nov 1, 2014 at 9:05 vote accept user35593
Oct 29, 2014 at 1:13 history edited Anton Petrunin CC BY-SA 3.0
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Oct 29, 2014 at 1:04 comment added Anton Petrunin I did not see it, but the same construction gives non-uniqueness for $m=3$ and $n=4$; I will update my answer.
Oct 28, 2014 at 5:42 comment added user35593 Yes but then $a_1,\dots,a_m$ do all lie on the same geodesic (The geodesic connecting $a_1$ and $a_2$). My claim excludes this case.
Oct 28, 2014 at 1:48 history answered Anton Petrunin CC BY-SA 3.0