Menger curvature and rectifiability

Abstract

For a Borel set E in R^n, the total Menger curvature of E, or c(E), is the integral over E^3 (with respect to 1-dimensional Hausdorff measure in each factor of E) of c(x,y,z)^2, where 1/c(x,y,z) is the radius of the circle passing through three points x, y, and z in E. Let H^1(X) denote the 1-dimensional Hausdorff measure of a set X. A Borel set E in R^n is purely unrectifiable if for any Lipschitz function gamma from R to R^n, H^1(E cap gamma(R)) = 0. It is said to be rectifiable if there exists a countable family of Lipschitz functions gamma_i from R to R^n such that H^1(E - union gamma_i(R)) = 0. It may be seen from this definition that any 1-set E (that is, E Borel and 0<H^1(E)<\infty) can be decomposed into two disjoint subsets E_irr and E_rect, where E_irr is purely unrectifiable and E_rect is rectifiable. Theorem. If E is a 1-set in R^n and c(E)^2 is finite, then E is rectifiable.