A notion of continuity in discrete spaces and applications

TL;DR

This paper introduces a new concept of continuity for discrete spaces, enabling the extension of classical theorems and inequalities from finite graphs to broader classes of locally finite metric spaces.

Contribution

It defines a notion of continuous paths in discrete spaces and applies it to generalize the Jordan curve theorem and ^p-distortion inequalities.

Findings

Established a discrete analogue of the Jordan curve theorem.

Extended ^p-distortion inequalities to a wide class of locally finite metric spaces.

Provided a framework connecting graph theory and metric space analysis.

Abstract

We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z^2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the \ell^p-distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.