Invariants for Gromov's pyramids and their applications

TL;DR

This paper develops a theory of invariants for Gromov's pyramids, which are generalized metric measure spaces, and demonstrates their ability to distinguish pyramids and analyze the structure of the space of pyramids.

Contribution

It introduces a general framework for invariants of pyramids, constructs multiple invariants, and proves the infinite-dimensionality of the space of non-trivial pyramids.

Findings

Constructed several invariants for pyramids.

Distinguished specific pyramids using these invariants.

Proved the space of non-trivial pyramids is infinite-dimensional.

Abstract

Pyramids introduced by Gromov are generalized objects of metric spaces with Borel probability measures. We study non-trivial pyramids, where non-trivial means that they are not represented as metric measure spaces. In this paper, we establish general theory of invariants of pyramids and construct several invariants. Using them, we distinguish concrete pyramids. Furthermore, we study a space consisting of non-trivial pyramids and prove that the space have infinite dimension.