Lecture 3: 3N colored points in a plane

Thu, 24/03/201116:00
Prof. Günter M. Ziegler, Freie Universität Berlin
More than 50 years ago, the Cambridge undergraduate Bryan Birch showed that any 3N points in a plane can be split into N triples that span triangles with a non-empty intersection. He also conjectured a sharp, higher-dimensional version of this, which was proved by Helge Tverberg in 1964 (freezing, in a hotel room in Manchester).

In a 1988 Computational Geometry paper, Bárány, Füredi & Lovász noted that they needed a colored version of Tverberg's theorem. Bárány & Larman proved such a theorem for 3N colored points in a plane, and conjectured a version for d dimensions. A remarkable 1992 paper by Zivaljevic & Vrecica obtained such a result, though not with a tight bound on the number of points.

We now propose a new colored Tverberg theorem, which is tight, which generalizes Tverberg's original theorem - and which has three quite different proofs. Pick your favourite!
(Joint work with Pavle V. Blagojevic and Benjamin Matschke)