In Theorem 26.5, we have proved the C1,γ-regularity of the reduced boundaries A∩E of any (Λ, r0)-perimeter minimizers E in some open set A. This chapter is devoted to the study of the singular set

aiming to provide estimates on its possible size. From the density estimates of Theorem 21.11, we already know that ℋn−1(Σ(E; A)) = 0. This result can be largely strengthened, as explained in the following theorem.

Theorem 28.1 (Dimensional estimates of singular sets of (Λ, r0)-perimeter minimizers) If E is a (Λ, r0)-perimeter minimizer in the open set A ⊂ ℝn, n ≥ 2, with Λr0 ≤ 1, then the following statements hold true:

1. (i) if 2 ≤ n ≤ 7, then Σ(E; A) is empty;

2. (ii) if n = 8, then Σ(E; A) has no accumulation points in A;

3. (iii) if n ≥ 9, then ℋs(Σ(E; A)) = 0 for every s > n − 8.

There exists a perimeter minimizer E in ℝ8with H0(Σ(E;∈8)) = 1. If n ≥ 9, then there exists a perimeter minimizer E in ℝn with Hn−8(Σ(E;ℝn)) = ∞.

Let us now sketch the proof of this deep theorem. We start by looking at the blow-ups Ex, r of a (Λ, r0)-minimizer E at a singular point x. In Theorem 28.6, we show that, loosely speaking, the Ex, r converge to a cone K with vertex at 0.