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Tverberg plus minus

Connections for Women Workshop: Geometric and Topological Combinatorics August 31, 2017 - September 01, 2017

September 01, 2017 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Imre Barany (Alfréd Rényi Institute of Mathematics)
Location: MSRI: Simons Auditorium
  • Tverberg's theorem

  • sign conditions

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



We prove a Tverberg type theorem: Given a set $A \subset \R^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ (where $|A_p|\le d+1$ for each $p$) with the following property. The unique $z \in \bigcap_{p=1}^r \aff A_p$ can be written as an affine combination of the elements in $A_p$: $z=\sum_{x\in A_p}\al(x)x$ for every $p$ and exactly $k$ of the coefficients $\al(x)$ are negative. The case $k=0$ is Tverberg's classical theorem. This is joint works with Pablo Soberon.

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