##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Boris Bukh

#### Princeton University

## Stabbing simplices by points and affine spaces

##### Abstract:

B\'ar\'any showed that there is a constant $c_d>0$ such that if $S$ is any $n$-point set in $R^d$, then there exists a point in $c_d$ fraction of simplices spanned by $S$. We present a simple construction of a point set for which there is no point contained in many simplices. The construction is optimal for $d=2$ and gives the first non-trivial upper bounds on $c_d$ for $d\geq 3$. We will also discuss generalizations to stabbing simplices by affine spaces. Joint work with Ji\v{r}\'\i{} Matou\v{s}ek and Gabriel Nivasch.

Host: Jacques Verstraete

### March 25, 2008

### 4:00 PM

### AP&M 7321

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