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Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood.

Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$?

Comments.

  • If $\dim H=2$, then the answer is yes.

  • I can show that $\partial \mathrm{Conv}\, F\cap \partial \mathrm{Conv}\, W$ cannot have isolated points.

  • An analogous statement for CAT(0) metrics on the plane does not hold.

  • This question is motivated by the last lines in this question

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  • $\begingroup$ Is it easy to describe the CAT(0) metrics on the plane that provide counterexamples to this? $\endgroup$
    – Andy Putman
    Commented Feb 5 at 2:40
  • $\begingroup$ @AndyPutman take flat plane with one singular point $p$ with total angle $>2\cdot\pi$ arount it. If $p\in\partial(\mathrm{Conv}\,F)$, then likely $p\in\partial(\mathrm{Conv}\,W)$ $\endgroup$
    – Anton Petrunin
    Commented Feb 5 at 3:04
  • $\begingroup$ That makes sense, thanks! $\endgroup$
    – Andy Putman
    Commented Feb 5 at 12:25

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