Bounds for α-optimal partitioning of a measurable space based on several efficient partitions

Abstract

We provide a two-sided inequality for the α-optimal partition value of a measurable space according to a finite number of nonatomic finite measures. The result extends and often improves Legut [Inequalities for α-optimal partitioning of a measurable space, Proc. Amer. Math. Soc. 104 (1988)] since the bounds are obtained considering several partitions that maximize the weighted sum of the partition values with varying weights, instead of a single one. Furthermore, we show conditions that make these bounds sharper.