A note on the polar decomposition in metric spaces

TL;DR

This paper discusses the technical aspects of defining polar decompositions in metric spaces, which can aid in integrating over these spaces with regular measures.

Contribution

It provides a detailed analysis of the conditions and technical details necessary for establishing polar decompositions in metric spaces.

Findings

Clarifies the conditions for well-defined polar decompositions in metric spaces

Addresses technical challenges in extending Euclidean polar coordinates to metric spaces

Lays groundwork for future applications in integration theory in metric spaces

Abstract

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions.