As explained in Remark 23.8, the Lipschitz approximation theorem reduces the regularity problem by showing that, by perimeter minimality, the smallness of the excess at a given scale implies the smallness of the excess at every smaller scale. To prove this, in Chapter 25, we shall need a reverse height bound, in which the excess is controlled through a sort of L2-height. Precisely, we introduce the cylindrical flatness of a set of locally finite perimeter E ⊂ ℝn at x ∈ ℝn with respect to υ ∈ Sn−1 at scale r > 0, as

The flatness f(E, x, r, υ) measures the L2-average distance of E from the family of hyperplanes {y: (y − x) · υ = c} (c ∈ ℝ) in the cylinder C(x, r, υ). We now provide the required bound (24.2) on the excess in terms of the flatness. In the statement, ω(n, t0) denotes the constant introduced in Lemma 22.10.

Theorem 24.1 (Reverse Poincaré inequality) There exists a positive constant C(n) with the following property. If E is a (Λ, r0)-perimeter minimizer inC(x0, 4r, υ) with

and with

then

Remark 24.2 For technical reasons, the proof of this result is a bit lengthy. However, since it contains no ideas which are going to be reused in other parts of the book, we strongly suggest skipping it on a first reading, directly moving to Chapter 25.