Hyperconvex metric spaces - a case study by Harry Petyt

I work in the field of geometric group theory. This is a pretty broad heading, but for me it means that the goal is to understand infinite groups, and the strategy is to get them to act on nice metric spaces that we know a lot about. 

For this strategy to be viable, we need to actually have classes of metric spaces that we understand well and that are suitable for finding group actions. There are quite a few such classes, but here I'll focus on one in particular that I've been interested in lately: the class of hyperconvex metric spaces, which dates back to the 1950s [AP56]. Intuitively, hyperconvexity means that every collection of pairwise-intersecting closed balls has nonempty total intersection. 

Definition.  A metric space is hyperconvex if for every collection $\{B(x_i,r_i)\}$ of closed balls such that $\mathsf{d}(x_i,x_j)\le r_i+r_j$ for all $i,j$, there is a point that lies in all balls in the collection.

Here are some fundamental examples and non-examples.
$\checkmark$  The real line $\mathbf R$. More generally, any tree.
$\times$  Euclidean spaces of dimension greater than one are not hyperconvex, as can be seen by taking three discs in the plane that overlap pairwise but don't have a point in common.
$\checkmark$  The plane with the $\ell^\infty$ metric, i.e. the metric coming from the sup-norm. Here, balls are squares whose sides are parallel to the axes.
$\checkmark$  More generally, $\mathbf{R}^n$ with the $\ell^\infty$ metric, or indeed any product of hyperconvex spaces, together with the $\ell^\infty$ metric.
$\checkmark$  The plane with the $\ell^1$, or taxicab, metric. Balls are diamonds, so this is essentially just a rotated version of the plane with the $\ell^\infty$ metric.
$\times$  $\mathbf{R}^3$ with the $\ell^1$ metric is not hyperconvex. Every collection of three pairwise-intersecting closed balls has nonempty total intersection, but there are sets of four pairwise-intersecting balls with no point in common.
$\times$  Disconnected metric spaces are not hyperconvex.
$\checkmark$  Many simplicial complexes can be made hyperconvex by putting the $\ell^\infty$ metric on each cell [Hae22].
$\checkmark$  Surprisingly, it turns out that every metric space can be isometrically embedded in a unique  "smallest" hyperconvex space [Isb64].
$\checkmark$  There is even a characterisation of which subsets of $\mathbf{R}^n$ are hyperconvex with respect to the subspace $\ell^\infty$ metric [DP17].

There has been quite a bit of interest in these spaces in geometric group theory recently, for example in [Lan13, HO21, CCGHO20, HHP23, Hae24, SZ22], but I'm not going to get into any of that here. I'd just like to discuss a nice stand-alone problem that I think doesn't really require any context.

The basic issue is this. If I give you a metric space and ask you whether it is hyperconvex, then in principle you will have to go and look at all collections of balls of all possible sizes to be able to answer the question. In particular, you have to look at every single subset of the space. That might be difficult to do in practice. What we would like is a sufficient condition for hyperconvexity that is less taxing to check. For instance:

Question 1.  Under what conditions on a metric space $X$ can you tell whether it is hyperconvex by looking at only finitely many points of $X$ at once?

Let's introduce some terminology to make this question easier to discuss.

Definition.  A collection $\{B(x_i,r_i)\}$ of closed balls in a metric space $X$ is deep if $\mathsf{d}(x_i,x_j)\le r_i+r_j$ for all $i,j$. Say that $X$ is 
$*$  $n$-hyperconvex if every deep, cardinality-$n$ collection of balls has nonempty total intersection;
$*$  finitely hyperconvex if every deep, finite collection of balls has nonempty total intersection;
$*$  countably hyperconvex if every deep, countable collection of balls has nonempty total intersection. 

We have already seen in the above example of $\mathbf R^3$ with the $\ell^1$ metric that even an extremely nice 3-hyperconvex space can fail to be 4-hyperconvex. So we should resign ourselves to checking more than three points at once when we try to answer Question 1.

Another observation is that countable hyperconvexity of $X$ implies that $X$ is complete, because if we are given a Cauchy sequence then we can take a corresponding sequence of increasingly small balls, and countable hyperconvexity provides a point in the total intersection, which must be a limit of the sequence. We should therefore require $X$ to be complete in Question 1.

A more subtle example shows that finite hyperconvexity does not imply countable hyperconvexity in general, even for complete spaces. Let $X$ be the subspace of the space $\ell^\infty(\mathbf{N})$ of bounded sequences that consists of all sequences that are eventually zero. The space $X$ is finitely hyperconvex, but is not complete, so it cannot be countably hyperconvex. More interestingly, the completion $\bar X$ of $X$ is finitely hyperconvex [MP23], but it is also not countably hyperconvex, which can be seen as follows. Let $x^n=(x^n_i)$ denote the sequence such that $x^n_i=0$ if $i\ge n$ and $x^n_i=2$ if $i<n$. If a sequence $z$ lies in the intersection of the unit ball centred on $x^n$ with the unit ball centred on $x^1$, which is the zero sequence, then the first $n-1$ terms of $z$ are all 1. Thus the unit balls centred on the $x^n$ intersect pairwise but have empty total intersection in $\bar X$, because the sequence whose terms are all 1 lies at distance 1 from $X$.

The essential issue in this last example is that the space we are considering is infinite-dimensional. To sum up, we have seen that any answer to Question 1 should at least assume that $X$ is complete and finite-dimensional, and at a minimum we will need to check all sets of four points. With Thomas Haettel and Nima Hoda, we proved that these assumptions suffice [HHP].

Theorem.  Let $X$ be a complete metric space of finite topological dimension. If $X$ is 4-hyperconvex, then $X$ is hyperconvex.

Harry Petyt is a Titchmarsh Research Fellow here in Oxford Mathematics.

References:

[AP56] N. Aronszajn and P. Panitchpakdi. Extension of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math., 6:405–439, 1956.

[CCGHO20] Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, and Damian Osajda. Helly groups. arXiv:2002.06895, 2020.