Title

Optimal-Partitioning Inequalities for Nonatomic Probability Measures

Author Info

John Elton, Georgia Institute of Technology - Main Campus
Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow
Robert P. Kertz, Georgia Institute of Technology - Main Campus

Recommended Citation

Published in Transactions of the American Mathematical Society, Volume 296, Issue 2, August 1, 1986, pages 703-725. Copyright © 1986 Americn Mathematical Society. The definitive version is available at http://www.jstor.org/stable/2000385.

NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.

Abstract

Suppose μ1,...,μn are nonatomic probability measures on the same measurable space (S, B). Then there exists a measurable partition {Si}ⁿ_i=1 of S such that μi(Si) ≥ (n + 1 - M)^-1 for all i = 1,...,n, where M is the total mass of Vⁿ_i=1μ1 (the smallest measure majorizing each μi). 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 L1-functions, discrete allocation problems, statistical decision theory, and a dual problem.

Disciplines

Mathematics