Direct sums and decompositions of Gromov's pyramids

TL;DR

This paper introduces the concept of direct sums of Gromov's pyramids, proves their unique decomposition, and provides a method to determine if a pyramid is an extended metric measure space.

Contribution

It defines direct sums of pyramids, proves their unique decomposition, and offers a new way to identify extended metric measure spaces within pyramids.

Findings

Any pyramid admits a unique direct sum decomposition.

The method to check if a pyramid is an extended metric measure space.

Introduction of the direct sum concept for Gromov's pyramids.

Abstract

Gromov introduced the notion of a pyramid as a generalization of a metric measure space, based on the idea of the concentration of measure phenomenon. In this paper, we introduce the concept of a direct sum of pyramids, which naturally appears as a limit of a sequence of metric measure spaces whose measures concentrate on finitely or countably many regions, with the distances between these regions diverging to infinity. As one of our main results, we prove that any pyramid admits a unique direct sum decomposition. Moreover, as an application, we establish the method for checking whether a given pyramid is an extended metric measure space.