Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, December 06, 2011, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Gabriel Nivasch (EPFL)

For every fixed *d* and every *n*, we construct an *n*-point set *G* in **R**^{d} such that every line in **R**^{d} is contained in a halfspace that contains only 2*n*/(*d*+2) points of *G* (up to lower-order terms).

Apparently, the point set *G* satisfies the following more general property: For every *k*, every *k*-flat in **R**^{d} is contained in a halfspace that contains only (*k*+1) *n* / (*d*+*k*+1) points of *G* (up to lower-order terms).

In 2008, Bukh, Matousek, and Nivasch conjectured the following generalization of Rado's centerpoint theorem: For every *n*-point set *S* in **R**^{d} and every *k*, there exists a *k*-flat *f* in **R**^{d} such that every halfspace that contains *f* contains at least (*k*+1) *n* / (*d*+*k*+1) points of *S* (up to lower-order terms). (The centerpoint theorem is obtained by setting *k*=0.) Such a flat *f* would be called a "centerflat".

Thus, our upper bound construction shows that the leading constant (*k*+1)/(*k*+*d*+1) in the above conjecture is tight (certainly for *k* = 1, and apparently for all *k*).

The set *G* of our construction is the "stretched grid" -- a point set which has been previously used by Bukh et al. for other related purposes.

Joint work with Boris Bukh.

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