this is related to [this question](https://mathoverflow.net/questions/6627/convex-hull-in-cat0) but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be closed. I was wondering if anyone was aware of an explicit example ? 



Example of non-closed convex hull in a CAT(0) space
There are such examples already in Riemannian world!
In fact in any generic Riemannian manifold of dimension $\ge3$ convex hull of 3 points in general position is not closed.
BUT it is hard to make explicit and generic at the same time :)


If it is closed then there are a lot of geodesics lying in its boundary --- that is rare!
To see it do the following exercise first: *Show that in generic 3-dimensional manifold, arbitrary smooth convex surface contains no geodesic.* (Here geodesic = geodesic in ambient space.)


To make word "generic" more clear: show that any metric admits $C^\infty$-perturbation such that above property holds.


**Semisolution:**
Assume that a geodesic $\gamma$ lies in the boundary of a convex set $K$ with smooth boundary. Let $N(t)$ be the outer normal vector to $K$ at $\gamma(t)$. Note that $N(t)$ is parallel.
Further note that from convexity of $K$ we get that for any Jacoby field $J(t)$ such that
$$\langle N(t\_0),J(t\_0)\rangle\le 0\ \text{and}\ \langle N(t\_1),J(t\_1)\rangle\le 0,$$
we have
$$\langle N(t),J(t)\rangle\le 0\ \text{if}\ t\_0<t<t\_1.$$
Note that this condition does not hold if the curvature tensor on $\gamma$ is generic.


**P.S.** Roughly it means that convex hulls in Riemannian world are too complicated. But I know one example where it is used, see Kleiner's *An isoperimetric comparison theorem*.
But he is only using that Gauss curvature of non-extremal points on the boundary of convex hulls is zero...


**Appendix.** *(A construction of convex hull.)* To construct convex hull you can do the following: start with some set $K\_0$ and construct a sequence of sets $K\_n$ so that $K\_{n+1}$ is a union of all geodesics with ends in $K\_n$. The union $W$ of all $K\_n$ is convex hull. Now assume it coincides with its closure $\bar w$. In particular if $x\in\partial\bar W$ then $x\in K\_n$ for some $n$. I.e. there is a geodesic in $\bar W$ passing through $x$ (if $x\not\in K\_0$). From convexity, it is clear that such geodesic lies in $\partial \bar W$...


**P.P.S.** A more general statement is proved in our paper [*About every convex set...*](https://arxiv.org/abs/2103.15189)


