Proof. Suppose ∂ Ω has a cusp at γ ( s ). Then, using the terminology of definition 4.2.4 and the fact that γ is arc length parameterized, we have Γ + = − Γ $^{−}$. We let w = 0 and note that w = Γ + + Γ − ∈ T$_{γ}$ ( s ). Letting u, v ∈ ∂ Ω ∩ D ( γ ( s ) , r ) with u = v , we have 〈 w, u − v 〉 = 0, contradicting the fact that ∂ Ω is tangent-cone graph-like. Therefore, ∂ Ω has no cusps. The following technical lemmas allow us to bound various distances and areas encoun- tered in tangent-cone graph-like boundaries. Lemma 4.2.16. Suppose that ∂ Ω is tangent-cone graph-like with radius r and points p$_{1}$, p$_{2}$ ∈ ∂ Ω with d ( p$_{1}$, p$_{2}$ ) < r . Then one of the arcs (call it P ) along ∂ Ω between p$_{1}$ and p$_{2}$ is such that, for any two points q$_{1}$, q$_{2}$ ∈ P , we have d ( q$_{1}$, q$_{2}$ ) < r . Proof. Note that p$_{2}$ ∈ D ( p$_{1}$, r ) so that there is an arc along ∂ Ω from p$_{1}$ to p$_{2}$ which is fully contained in the interior of D ( p$_{1}$, r ) by theorem 4.2.1. We will call this arc P . For all x on P , let P$_{x}$ denote the subpath of P from p$_{1}$ to x (so P = P$_{p}$$_{2}$ ). We claim that P$_{x}$ is contained in D ( x, r ) for all x on P (thus, P is contained in D ( p$_{2}$, r )). Indeed, if this were not the case, then there must be some ˆ x on P such that P$_{ˆ}$ x is contained in D (ˆ x, r ) but C (ˆ x, r ) ∩ P$_{ˆ}$ x is nonempty (i.e., we can move the disk along P until some part of the subpath hits the boundary). That is, the subpath P$_{ˆ}$ x has a tangency with the disk D (ˆ x, r ) which is impossible because of theorem 4.2.1. Let q$_{1}$ ∈ P and note that since P$_{x}$ is contained in D ( x, r ) for all x on P , we have that P is contained in D ( q$_{1}$, r ). Therefore, d ( q$_{1}$, q$_{2}$ ) < r for all q$_{1}$, q$_{2}$ ∈ P as desired. Lemma 4.2.17. If q$_{1}$ = γ ( s$_{1}$ ) , q$_{2}$ = γ ( s$_{2}$ ) ∈ P where P is as in the previous lemma, then the arc length between q$_{1}$ and q$_{2}$ along P is at most √$_{2}$$_{d}$$_{(}$$_{q}$$_{1}$$_{, q}$$_{2}$$_{)}$$_{.}$ 94 4.2.5 TCGL Boundary Properties