Let us now briefly sketch a picture of the situation after the results proved in the previous chapters. For the sake of clarity, in this overview, we shall directly refer to the case of a perimeter minimizer E in an open cylinder C(x0, r0), with x0 ∈ E (thus, Λ = 0). In the Lipschitz approximation theorem, Theorem 23.7, we have shown the existence of two constants ε1(n) and δ0(n) such that if e(E, x0, 9 r, υ) ≤ ε1(n) and 9r < r0, then there exists a Lipschitz function u: ℝn−1 → ℝ, which is “almost harmonic” on D(px0, r), and whose graph covers the “good” part M0 of C(x0, r, υ) ∩ E,

In turn, we were able to show that M0 covers a portion of C(x0, r, υ)∩E which is large in proportion to the smallness of e(E, x0, 9 r, υ). However, our final goal is showing that the two sets coincide: that is to say, we would like to show that, provided e(E, x0, 9 r, υ) is small enough, then one actually has

for every x ∈ C(x0, r, υ) ∩ E and s ∈ (0, 8 r). Clearly, this would imply that C(x0, r, υ) ∩ E coincides with the graph of u over D(px0, r).