Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

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

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Gabriel Nivasch (EPFL)

Upper bounds for centerflats

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 2n/(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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