TITLE: Tverberg's theorem in CAT(0) spaces
QUESTION [12 upvotes]: 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 $E_i$?

REPLY [14 votes]: 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.