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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.

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But it is okey for 4 points... –  Anton Petrunin Nov 29 '13 at 23:57
    
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
    
So, this might parallel the situation with "topological Tverberg's theorem". –  Misha Nov 30 '13 at 7:15
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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
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non-manifold case is different. However, it would be interesting to figure out what the right bound is in the pure CAT(0) case. At the moment, I do not have a conjecture. Here is the reference to topological Tverberg: renyi.hu/~barany/cikkek/5.pdf. –  Misha Dec 1 '13 at 21:32
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