On Helly's theorem in geodesic spaces

In this short note we show that Helly's Intersection Theorem holds for convex sets in uniquely geodesic spaces (in particular in CAT(0) spaces) without the assumption that the convex sets are open or closed.


Introduction
The classic Helly's Intersecton Theorem asserts the following: If {A i } is a finite collection of convex sets in R n such that every subcollection consisting of at most n + 1 sets has a nonempty intersection, then A i = ∅. This theorem has a topological generalization (found by Helly himself [6]) where convexity is replaced by the assumption that the sets A i and their nonempty intersections are open homology cells. See [4] for a modern proof and further references.
The proof of the topological Helly's theorem extends to CAT(0) spaces of geometric dimension n, see e.g. [8,Proposition 5.3] and [5, §3]. Thus Helly's theorem holds for open convex sets in such spaces. Once the theorem is established for open sets, the variant with closed convex sets follows. In R n , one can deduce the theorem for arbitrary convex sets by picking one point in every nonempty intersection and replacing every set by the convex hull of the marked points it contains. However this argument does not work in CAT(0) spaces since convex hulls of finite sets are not necessarily closed.
In this note we show that Helly's theorem holds for arbitrary (not necessarily open or closed) convex sets in CAT(0) and some other spaces. Namely we prove the following. Theorem 1.1. Let X be a uniquely geodesic space of compact topological dimension n < ∞. Let {A i } be a finite collection of convex sets in X such that every subcollection of cardinality at most n + 1 has a nonempty intersection. Then A i = ∅.

Definitions.
Here are the definitions of terms used in Theorem 1.1.
A geodesic space is a metric space X such that every two points in X belong to a segment, where a segment is a subset isometric to a compact interval of the real line. We say that X is uniquely geodesic if for every x, y ∈ X there is a unique segment [xy] ⊂ X with endpoints at x and y, and [xy] depends continuously on x and y. Note that the continuous dependence is automatic if X is proper (i.e., if all closed balls are compact).
where dim is the Lebesgue covering dimension. For (locally) CAT(κ) spaces, the compact topological dimension equals the geometric dimension and a number of other dimension-like quantities [8].
The proof of Theorem 1.1 is topological, it uses only contractibility of convex sets. See Proposition 2.2 for a purely topological formulation. However, unlike the above mentioned proof of the topological Helly's theorem, the proof of Proposition 2.
is an open covering of ∆ such that G i ∩ F i = ∅ for each i, then G i = ∅. This fact is a topological variant of Sperner's lemma and follows easily from the discrete counterpart. Alternatively, it follows from the Knaster-Kuratowski-Mazurkiewicz lemma [9] which is a slightly more general statement about open or closed coverings of the simplex. Now proceed with the proof of Lemma 2.1. We may assume that X is compact, otherwise take f (∆) for X. Then dim X = dim c X ≤ n. Suppose, towards a contradiction, that f (F i ) = ∅. Then the sets U i = X \ f (F i ) form an open covering of X. By the definition of the covering dimension, there exists an open covering {V j } j∈J refining {U i } and having covering multiplicity at most n + 1. Let U ′ i be the union of those sets V j that are contained in U i but not in U 1 , . . . , U i−1 . Since the covering multiplicity of {V j } is less than n + 2, we have ) satisfy the assumptions of the topological Sperner's lemma above. Applying the lemma yields that f −1 (U ′ i ) = ∅ and hence U ′ i = ∅, a contradiction. Proposition 2.2. Let X be a contractible Hausdorff space with dim c X = n < ∞.
be a finite collection of contractible sets in X such that the intersection of every subcollection is either contractible or empty. Suppose that m ≥ n + 2 and for every set I ⊂ [m] with |I| = n + 1 one has i∈I A i = ∅. Then Then extend the map by induction as follows. Assuming that f is already defined on the (k − 1)-skeleton of ∆, where 1 ≤ k ≤ n + 1, consider a k-dimensional face ∆ I . Observe that f (∂∆ I ) ⊂ P I . Indeed, ∂∆ I = i∈I ∆ I\{i} and for every i ∈ I one has f (∆ I\{i} ) ⊂ P I\{i} = P I ∩ A i . Since P I is contractible, f | ∂∆I extends to a map from ∆ I to P I . Applying this extension procedure to all k-dimensional faces for k = 1, 2, . . . , n + 1, one gets the desired map f : ∆ → X.
This completes the proof in the case m = n + 2. The general case follows by induction in m. Let m > n + 2 and a collection {A i } m i=1 satisfy the assumptions of the proposition. Then, since the case m = n + 2 is already done, every subcollection of cardinality n+2 has a nonempty intersection. Therefore the collection {A ′ i } m−1 i=1 where A ′ i = A i ∩ A m satisfies the assumptions as well. Applying the induction hypothesis to {A ′ i } yields that the intersection A i is nonempty. In a uniquely geodesic space convex sets are contractible. This is ensured by the requirement that segments depend continuously on their endpoints. Intersections of convex sets are obviously convex and hence contractible. Therefore Theorem 1.1 follows from Proposition 2.2.