Does Helly's theorem hold in the hyperbolic plane?

Question

The Helly theorem in the Euclidean plane asserts that if $S_1, \dots, S_n$ are $n \ge 3$ convex subsets such that $S_i \cap S_j \cap S_k \ne \emptyset$ for all distinct triples $i,j,k$, then the total intersection $\bigcap_{i = 1}^n S_i$ is also nonempty.

I'm wondering if the same theorem is true in the hyperbolic plane (for concreteness, let's assume the Poincaré disk model). My understanding is that if the analogue of Radon's theorem is true in this setting, then Helly follows axiomatically.

Radon's theorem in the Euclidean plane asserts that given any four points $x_1, \dots, x_4$, there is a partition into two nonempty subsets such that the convex hulls intersect. The proof I know uses the affine structure on the Euclidean plane and so doesn't seem to port directly into hyperbolic space. On the other hand, I can verify that the Radon property holds for all the collections of four points I've looked at...

Answer 1

I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative statement about convex sets in the Euclidean plane holds in the Hyperbolic plane and vice versa, since the model maps convex sets to convex sets.

EDIT This has (almost) absolutely nothing to do with the above, but the topological version of Helly's theorem goes back to at least Debrunner (and it is a Monthly paper, so is human-readable), no need to allude to Farb's paper.

Answer 2

The original proof of Helly's theorem was topological and only uses basic homological properties of convex sets. It generalizes to all sorts of contexts, including the one you are interested in. Here is a general statement of what it can do. A homology cell is a topological space whose reduced singular homology is the same as that of a point (this implies in particular that it is nonempty).

Theorem: Let $X$ be a normal topological space such that for some $n \geq 1$, every open set $Y \subset X$ satisfies $H_q(Y)=0$ for $q \geq n$. Let $X_1,\ldots,X_k$ be a collection of closed homology cells in $X$. Assume that the intersection of any $r$ of the $X_i$ is nonempty for all $r \leq n+1$ and is a homology cell for $r \leq n$. Then the intersection of all the $X_i$ is a homology cell (and in particular is nonempty).

A discussion of this with references is in Section 3 of

B. Farb, Group actions and Helly's theorem, Adv. Math. 222 (2009), no. 5, 1574–1588.