4.2 Notation and Preliminaries 81 Θ $^{m}$( A, p ) = lim r ↓ 0 H $^{m}$( A ∩ D ( p, r )) α$_{m}$r m Figure 4.1: Notation and basic setup Unless otherwise specified, we will be assuming throughout this paper that Ω ⊂ R 2 is a compact set with simple closed, piecewise continuously differentiable boundary ∂ Ω of length L . Let γ : [0 , L ] → ∂ Ω be a continuous arclength parameterization of ∂ Ω (see Figure 4.1). We will adopt the convention that γ traverses ∂ Ω in a counterclockwise direction so it always keeps the interior of Ω on the left (there is no compelling reason for this particular choice, but adopting a consistent convention allows us to avoid some ambiguities later). Note that γ (0) = γ ( L ) and that γ restricted to [0 , L ) is a bijection. Denote by D ( p, r ) the closed disk and C ( p, r ) the circle of radius r centered at the point p ∈ R $^{2}$. In geometric measure theory, the m -dimensional density of a set A ⊆ R n at a point p ∈ R n is given by GLYPH<0>GLYPH<0> GLYPH<1>GLYPH<1> GLYPH<0>GLYPH<0> GLYPH<0>GLYPH<0> GLYPH<1>GLYPH<1> GLYPH<1>GLYPH<1> Ω γ ≡ ∂ Ω r γ ( s )