## 4.2.2 Two-Arc Property 

The graph-like condition implies (in proof of the following lemma) that Ω will never be entirely contained in the disk, no matter where on the boundary we center it. That is, some part of Ω lies outside of D ( p, r ) for every p ∈ ∂ Ω. 

Lemma 4.2.8. Let r ∈ R + and p ∈ ∂ Ω . If ∂ Ω is graph-like on D ( p, r ) , then | ∂ Ω ∩ C ( p, r ) | ≥ 2 . 

Proof. Suppose by way of contradiction that | ∂ Ω ∩ C ( p, r ) | < 2. Since ∂ Ω is a simple closed curve, we have ∂ Ω ⊆ D ( p, r ). As ∂ Ω is graph-like at p with radius r , there exists some orientation for which ∂ Ω ∩ D ( p, r ) = ∂ Ω is the graph of a well-defined function. However, ∂ Ω is a simple closed curve so it is not the graph of a function in any orientation, yielding a contradiction. 

The next result is the reason we find the tangent-cone graph-like condition useful. It says that if ∂ Ω is tangent-cone graph-like with radius r , then, for every p ∈ ∂ Ω, the disk D ( p, r ) has only two points of intersection with ∂ Ω and these are transverse. In other words, this means that when working locally in the disk D ( p, r ) we need only consider a single piece of ∂ Ω. 

Theorem 4.2.1. If ∂ Ω is tangent-cone graph-like with radius r ∈ R + at p ∈ ∂ Ω , then | ∂ Ω ∩ C ( p, r ) | = 2 and ∂ Ω crosses C ( p, r ) transversely at these points. As a result, for every q$_{1}$, q$_{2}$ ∈ ∂ Ω ∩ D ( p, r ) , there is a unique arc along ∂ Ω between them in D ( p, r ) . 

Proof. By Lemma 4.2.8, we have that | ∂ Ω ∩ C ( p, r ) | ≥ 2. Note that ∂ Ω contains an interior point ( p ) and at least two boundary points of the disk D ( p, r ) (since | ∂ Ω ∩ C ( p, r ) | ≥ 2). As ∂ Ω is connected and simply closed, there must exist an arc of ∂ Ω within the disk going from some point on C ( p, r ) through p to another point on C ( p, r ). 

