Menger curvature and rectifiability in metric spaces

TL;DR

This paper establishes that a finite integral of Menger curvature over a metric space implies the space is rectifiable, linking geometric curvature measures to the structure of the space.

Contribution

It proves a new criterion connecting Menger curvature integrability with rectifiability in metric spaces, extending geometric analysis tools.

Findings

Finite Menger curvature integral implies rectifiability

Provides a curvature-based criterion for space structure

Extends geometric measure theory in metric spaces

Abstract

We show that for any metric space $X$ the condition \[ \int_X\int_X\int_X c(z_1,z_2,z_3)^2\, d\Hm z_1\, d\Hm z_2\, d\Hm z_3 < \infty, \] where $c(z_1,z_2,z_3)$ is the Menger curvature of the triple $(z_1,z_2,z_3)$, guarantees that $X$ is rectifiable.