TITLE: Examples of metric spaces with measurable midpoints
QUESTION [11 upvotes]: Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ such that $d(x,m(x,x')) = d(x',m(x,x')) = d(x,x') / 2$ for all $x,x' \in X$.
It seems to be known (e.g see section 6 of this paper) that continuous midpoint spaces (i.e Polish spaces with the continuous midpoint property) include:

Hilbert spaces.
Closed convex subsets of Banach spaces.
Hyperconvex spaces.
CAT(0) spaces.

Hopefully, the collection of measurable midpoint spaces contains much more general examples (for the above list is quite restrictive).

Question. What are some examples of measurable midpoint spaces ?

REPLY [2 votes]: We are ultimately looking at the complexity of the multivalued function $\mathrm{MidPoint}_\mathbf{X} : \mathbf{X} \times \mathbf{X} \rightrightarrows \mathbf{X}$ assigning some midpoint to the points here. This includes what choice functions there are, but need not be limited to it. The framework to study the complexity of such operations is Weihrauch reducibility.
Just by definition of the midpoint, it follows that the map from a pair of points to the closed set of midpoints (equipped with the lower Vietoris topology) is continuous. This tells us that $\mathrm{MidPoint}_\mathbf{X} \leq_{\mathrm{W}} \mathrm{C}_\mathbf{X}$, where $\mathrm{C}_\mathbf{X}$ is closed choice on $\mathbf{X}$, which maps non-empty closed sets to some element. With  $\mathrm{UC}_\mathbf{X}$ I denote the restriction of $\mathrm{C}_\mathbf{X}$ to singletons.
Everything we need about closed choice for this is found here.
Since we are assuming $\mathbf{X}$ to be Polish1, we immediately get the following:

If $\mathbf{X}$ is sigma-compact and midpoints are unique, then $\mathrm{MidPoint}_\mathbf{X} \leq_{\mathrm{W}} \mathrm{C}_\mathbb{N}$. This implies that the midpoint map is piecewise continuous, ie that there is a countable cover of $\mathbf{X} \times \mathbf{X}$ by closed sets, such that on each piece the map is continuous.
If $\mathbf{X}$ is sigma-compact, then $\mathrm{MidPoint}_\mathbf{X} \leq_{\mathrm{W}} \mathrm{C}_\mathbb{R}$. This guarantees that there is a Baire class 1 selection function for midpoints, but we get more [Baire class 1 is equivalent to "preimages of opens are $\Sigma^0_2$, so this is much simpler than Borel measurable]. For example, there always is a midpoint which is low (in the computability-theoretic sense) relative to the space.
If midpoints are unique, then $\mathrm{MidPoint}_\mathbf{X} \leq_{\mathrm{W}} \mathrm{UC}_{\mathbb{N}^\mathbb{N}}$. Since the domain of $\mathrm{MidPoint}_\mathbf{X}$ is a Polish space, this already implies that $\mathrm{MidPoint}_\mathbf{X}$ is Borel measurable.
Without any restrictions, we just get that $\mathrm{MidPoint}_\mathbf{X} \leq_{\mathrm{W}} \mathrm{C}_{\mathbb{N}^\mathbb{N}}$. This doesnt rule out that we could avoid Borel measurable selection functions for the midpoint, but any construction would need to be very weird. The best starting point I can think of is using diagonally non-arithmetic functions.

1 We don't need that the metric defining our midpoints is complete, we just need some equivalent complete metric around.