# Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$:

Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - 1) + 1$, there exists a point $x\in X$ and a partition of $E$ into $r$ subsets $E_1,...,E_r$, such that $x$ belongs to the intersection of closed convex hulls of the subsets $E_i$?

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No. Let $X$ be a tripod (three segments with one common endpoint), $d=1$, $r=2$ and $E$ the set of the 3 leaf points.
It seems that $2^{n+1}$ points in $n$-dimensional CAT(0) space can be always divided in two groups. – Anton Petrunin Nov 30 '13 at 0:08
In a Hadamard $d$-manifold, $d+2$ points seems to be enough for $r=2$. This follows by applying Borsuk-Ulam theorem to a suitable barycentric map. – Sergei Ivanov Dec 1 '13 at 9:53