We now introduce the notion of excess, a key concept in the regularity theory for (Λ, r0)-perimeter minimizers. Given a set of locally finite perimeter E in ℝn, the cylindrical excess of E at the point x ∈ E, at the scale r > 0, and with respect to the direction υ ∈ Sn−1, is defined as
see Figure 22.1. The spherical excess of E at the point x ∈ E and at scale r > 0 is similarly defined as
Hence, when considering the spherical excess at a given scale, we essentially minimize the cylindrical excess at that scale with respect to the direction. The fundamental result related to the notion of excess is that, if E is a (Λ, r0)- perimeter minimizer, then the smallness of e(E, x, r, υ) at some x ∈ E actually forces C(x, s, υ) ∩ E (for some s < r) to agree with the graph (with respect to the direction υ) of a C1,γ-function (see Theorem 26.1 for the case of local perimeter minimizers, and Theorem 26.3 for the general case). This theorem is proved through a long series of intermediate results, in which increasingly stronger conclusions are deduced from a small excess assumption. We begin this long journey in the next two chapters, where we shall prove, in particular, the so-called height bound, Theorem 22.8: if E is a (Λ, r0)-perimeter minimizer in C(x, 4r, υ) with x ∈ E and e(x, 4r, υ) suitably small, then the uniform distance of C(x, r, υ) ∩ E from the hyperplane passing through x and orthogonal to υ is bounded from above by e(x, 4r, υ)1/2(n−1).