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

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

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

Mittagsseminar Talk Information

Date and Time: Tuesday, March 04, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Marek Sulovský

Circle containment problem and centerpoints

The following problem was posed at last year's GWOP:

What is the maximum c such that for every point set S in the plane, one can find a pair of points p, q with the property, that every disc D containing both p and q contains at least a c-fraction of the whole S (i.e. D intersected with S has size at least c|S|). This problem also has a natural higher dimensional generalization.

This question was first asked by Urrutia and Neuman-Lara in 1988 and there was some research on it in 1988-1989 but since then it did not receive any attention.

We improve the constant c for the high dimensional problem and relate it to some questions concerning so called lower center regions. The main argument involves Clarkson-Shor random sampling method combined with some bounds on the numbers of k-sets.

Warning: Since we use many basic geometric concepts and geometry backgrounds of the mittagsseminarists varies a lot, I will rather assume no particular knowledge. This means that I will spend quite o lot of time explaining basics like centerpoints and lifting map - my apology to all of you familiar with these concepts.

Joint work with Shakhar Smorodinsky and Uli Wagner. Special thanks to Jozsef Solymosi and Patrick Traxler.

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