Rectifiable set

Summary

An $m$-rectifiable set is a subset of $\R^n$ covered by countably many Lipschitz images of $\R^m$. The cover may miss a set of $m$-dimensional Hausdorff measure zero. Such sets are the measure-theoretic version of $m$-dimensional surfaces, and they may have corners, cusps and countably many pieces. They are the basic objects of geometric measure theory, where they replace smooth submanifolds in problems such as the Plateau problem. Every set of finite $m$-dimensional Hausdorff measure splits into a rectifiable part and a purely unrectifiable part, and the two parts behave very differently.