MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $E_i$?

share|cite|improve this question
up vote 14 down vote accepted

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.

share|cite|improve this answer
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
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
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: – Misha Dec 1 '13 at 21:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.