Extension of Lyapunov's Convexity Theorem to Subranges

TL;DR

This paper extends Lyapunov's convexity theorem by demonstrating that the union of ranges of all subsets with a fixed vector measure remains compact and convex in atomless cases, with a geometric construction for these sets.

Contribution

It introduces a novel extension of Lyapunov's theorem to unions of subset ranges and provides a geometric construction for these convex compact sets.

Findings

Union of subset ranges is compact and convex in atomless measures.

Equality of the union and geometric convex set holds in 2D but can fail in higher dimensions.

Provides a geometric method to construct convex compacta containing the union.

Abstract

Consider a measurable space with a finite vector measure. This measure defines a mapping of the $\sigma$-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compactum in the Euclidean space that contains this union. The equality of these two sets, that holds for two-dimensional measures, can be violated in higher dimensions.