Example of non-closed convex hull in a CAT(0) space this is related to this question 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 ? 
 A: There is a natural example if you don't ask the collection of point to be finite (but where it is closed): take the subset $A_1$ of the real Hilbert space $X=L^2(\mathbb{R})$ consisting of function taking at most one value beside $0$. Then it is easily seen that the geodesic segments between pairs of points in $A_1$ cover the set $A_3$ of function taking at most $3$ values beside $0$. The convex hull of $A_1$ is then the set of functions taking a finite number of values, and is dense in $X$.
For the story, this is a variation on an example I came across in optimal transportation: $X$ was the Wasserstein space of the real line with quadratic cost, which is isometric to the subset of $L^2([0,1])$ consisting of non-decreasing functions.
A: 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...
