Optimal-partitioning inequalities for nonatomic probability measures

Abstract:

Suppose ${\mu _1}, \ldots ,{\mu _n}$ are nonatomic probability measures on the same measurable space $(S,\mathcal {B})$. Then there exists a measurable partition $\{ {S_i}\} _{i = 1}^n$ of $S$ such that ${\mu _i}({S_i}) \geq {(n + 1 - M)^{ - 1}}$ for all $i = 1, \ldots ,n$, where $M$ is the total mass of $\vee _{i = 1}^n {\mu _i}$ (the smallest measure majorizing each ${\mu _i}$). This inequality is the best possible for the functional $M$, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to ${L_1}$-functions, discrete allocation problems, statistical decision theory, and a dual problem.