Optimal-partitioning inequalities for nonatomic probability measures

Abstract

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