Guy David and Stephen Semmes
Quasiminimal Surfaces of Codimension 1 and John Domains

We study codimension 1 quasiminimizing surfaces in Rⁿ, and establish uniform rectifiability and other geometric properties of these surfaces. For instance, their complementary components must be John domains. In fact we give a complete characterization of quasiminimizers. As an application we show that sets which are not too large and which separate points in a definite way must have a large uniformly rectifiable piece. In this way we use area quasiminimizers to solve a problem in geometric measure theory.