# 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 ?

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 $$\langle N(t),J(t)\rangle\le 0\ \text{if}\ t_0 Note that this condition does not hold if the curvature tensor on $$\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...

• this is very helpful. However, there are some things that I'm not entirely clear about. * I'm not sure why in your statement about generic 3D manifolds, the geodesic has to be in the ambient space ? * I don't see why this implies the result (I'm not doubting: merely not following) – Suresh Venkat Dec 12 '09 at 6:34
• @Anton, I don't get it. In any manifold, take a point $p$ and a nearby small ball $B$, and connect $p$ to all points of $B$ by geodesic segments. I believe the resulting set is convex if everything is small enough, yet the conical part of the boundary is filled by geodesics. – Sergei Ivanov Dec 13 '13 at 17:49
• @SergeiIvanov, I extended the hint; now it is called "semisolution". I hope it helps. – Anton Petrunin Dec 14 '13 at 2:48
• Thanks, now I understand. Indeed the Euclidean intuition fails. – Sergei Ivanov Dec 14 '13 at 20:18

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.