Discrete Mathematics and Optimization Seminar
Wed, Oct. 13th, 2010
MC 320, 4:00 PM
Equipartitioning by convex k-fans
Imre Barany
Alfred Renyi Mathematical Institute
Abstract: A k-fan is a point in the plane and k halflines emanating from it. I'll explain a few results about equipartitions by k-fans of two or more probability measures, as well as partitions in other prescribed ratios. This group of questions is motivated by a neat problem of Kaneko and Kano from 1998. One of the results, which is joint with J Matousek, says that given two probability measures in the plane, there exists a 4-fan that simultaneously equipartitions them.

A recent question, raised by Nandakumar and Ramanda Rao, asks that, given a convex body C in the plane and a positive inetger k, is there a partition of C into k convex pieces so thate each piece has the same area and the same perimeter. I'll sketch the solution in the case k=3 which is a joint result with P Blagojevic and A Szucs. The methods use equivariant topology with a some extra geometry and combinatorics.