Simultaneous partitions of measures by k-fans

Abstract

A k-fan is a point in the plane and k semilines emanating from it. Motivated by a neat question of Kaneko and Kano, we study equipartitions by k-fans of two or more probability measures in the plane, as well as partitions in other prescribed ratios. One of our results is: for any two measures there is a 4-fan such that one of its sectors contains two-fifths of both measures, and each of the the remaining three sectors contains one-fifth of both measures.