From the Ham Sandwich to the Pizza Pie: A Simultaneous Z_m Equipartition of Complex Measures

Abstract

A "ham sandwich" theorem is derived for n complex Borel measures on C^n. For each integer m>=2, it shown that there exists a regular m-fan centered about a complex hyperplane, satisfying the condition that for each complex measure, the "Z_m rotational average" of the measures of the m corresponding regular sectors is zero. Taking [n/2] finite Borel measures on R^n and letting m=3, the theorem shows the existence of a regular 3-fan in R^n which trisects each measure; when m=4, the theorem shows the existence of a pair of orthogonal hyperplanes, each of which bisects each measure. If the theorem is applied to 2n finite Borel measures on R^2n, the classical ham sandwich theorem for R^2n is recovered when m = 2.