Given a Polish 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 ?

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 ?