A dual characterization of length spaces with application to Dirichlet metric spaces

TL;DR

This paper demonstrates that under minimal conditions, the intrinsic metric from a strongly local Dirichlet form creates a length space, using a dual characterization involving 1-Lipschitz functions forming a sheaf.

Contribution

It provides a dual characterization of length spaces and applies this to show that the intrinsic metric from a strongly local Dirichlet form induces a length space.

Findings

Intrinsic metric from Dirichlet form induces a length space

Dual characterization of length spaces via 1-Lipschitz functions

Application to Dirichlet metric spaces

Abstract

We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1-Lipschitz functions form a sheaf.