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Discrete Geometry on Colored Point Sets in the Plane

 Titel: Discrete Geometry on Colored Point Sets in the Plane Dozent(in): Prof. Dr. Mikio Kano, Ibaraki University, Ibaraki, Japan Termin: 28.05.2015, 16 Uhr s.t. Gebäude/Raum: Raum 2045, Multimediahörsaal, Gebäude N Ansprechpartner: Prof. Dr. Torben Hagerup

Inhalt:

Let R, B and G denote disjoint three sets of red points, blue points and green points in the plane (on a line), respectively. We discuss some problems on discrete geometry on R+B+G. Among some results, we show the following theorem, which is a generalization of Borusk-Ulam theorem. The hamburger theorem (Kano and Kyncl (2015, submitted)) If |R|+|B|+|G|=2n and 1 \le |R|, |B|, |G| \le n, then there exists a line l such that each open half plane H defined by l satisfies that (i) |(R+B+G)\cap H| \ge min {|R|,|B|,|G|}, (ii) |(R+B+G)\cap H| is even, (iii) 0 \le |R\cap H|, |B\cap H|,|G\cap H| \le |(R+B+G)\cap H|/2, and (iv) l passes through no point of R\cup B \cup G.