Checkerboards, Lipschitz functions and uniform rectifiability

Abstract

In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof. Theorem. Suppose O is a bounded open set in Rn with n > 2, and suppose that B(0,1) I O, Hn-1(?O) = M < 8 (depending on n and M) and a Lipschitz graph G (with constant L) such that Hn-1(G n ?O) = e. Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us to cover most of the unit sphere Sn-1.