A Cat(0) metric space $(X,d)$ of constant and finite local dimension is approximately flat if there exists a dense subset $U\subset X$ such that every $x\in U$ has a flat neighborhood (i.e. isometric to the euclidean space).
I'm interested in a generalization of Radon's theorem (https://matthewhr.wordpress.com/2013/02/27/radons-theorem/) to approximately flat Cat(0) spaces. Do you know if there exists one?
Thank you.
In dimension 1, Radon's theorem says that for any 3 points on the real line, one of them belongs to the segment between the two others. This becomes false if one replaces the real line by the tripod (the tree with 3 leaves), which is a 1-dimensional CAT(0) space. Indeed, none of the three leaf vertices of the tripod in on the segment between the two others.
So the most straightforward generalization fails. However if you are willing to increase the number of points in the set, then a generalization is possible. Moreover the local flatness in not needed.
Namely if $\dim X=n$ then $2n+2$ points is enough (I don't know if this bound is optimal). This follows from the case $r=2$ of Theorem 5.2 in this paper, which asserts that, for any continuous map $f:\Delta^{2n+1}\to X$, where $\Delta^{2n+1}$ is the standard $(2n+1)$-dimensional simplex, there exists two disjoint faces of the simplex whose images do intersect. Here $X$ is any $n$-dimensional metric space, not necessarily CAT(0).
To derive the Radon-type theorem from this fact it suffices to construct, for a given $S\subset X$ with $|S|=2n+2$, a map $f:\Delta^{2n+1}\to X$ which maps the vertices of the simplex to points of $S$ and maps each face of the simplex to the convex hull of the images of its vertices. This is easy to do in CAT(0).
