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Discrete Mathematics and Optimization Seminar
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Wed, Oct. 13th, 2010
MC 320, 4:00 PM
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Equipartitioning by convex k-fans
Imre Barany
Alfred Renyi Mathematical Institute
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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.
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