Bounds for alpha-Optimal Partitioning of a Measurable Space Based on Several Efficient Partitions

TL;DR

This paper introduces bounds for alpha-optimal partitions of a measurable space, extending previous results by considering multiple partitions that optimize weighted sums, leading to potentially tighter bounds.

Contribution

It extends Legut's 1988 results by deriving bounds based on multiple partitions that maximize weighted sums, offering a more comprehensive analysis.

Findings

Provides two-sided bounds for alpha-optimal partition values.

Extends previous work by considering multiple partitions with varying weights.

Improves bounds in many cases compared to earlier single-partition approaches.

Abstract

We provide a two-sided inequality for the alpha-optimal partition value of a measurable space according to n nonatomic finite measures. The result extends and often improves Legut (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.