23

The Lipschitz approximation theorem

The goal of this chapter is to prove that, if *E* is a (Λ, *r*_{0})-perimeter minimizer in **C**(*x*_{0}, 9*r*), with 9*r* < *r*_{0} and **e**_{n}(*x*_{0}, *r*) small enough, then **C**(*x*_{0}, *r*) ∩ *E* is almost entirely covered by the graph of a Lipschitz function *u*, which turns out to posses suitable almost-minimality properties (related to the (Λ, *r*_{0})-perimeter minimality of *E*). This is the content of the Lipschitz approximation theorem, Theorem 23.7, which is stated and proved in Section 23.3. Before coming to this, in Section 23.1 we discuss conditions under which the topological boundary of a set of finite perimeter *E* (normalized so that spt _{E}