since γ is TCGL at γ ( s ) with radius r and v$_{2}$ − v$_{1}$ ∈ T$_{γ}$ ( s ). Thus φ is TCGL at p with radius r − ϵ . The case where p = φ ( t ) is a vertex is similar but we must consider an arbitrary vector w ∈ T$_{φ}$ ( t ) in the inner product. We wish to show that, for every w ∈ T$_{φ}$ ( t ), there is a s ′ such that either w or − w ∈ T$_{γ}$ ( s $^{′}$) and d ( p, γ ( s $^{′}$)) ≤ ϵ $_{3}$, after which the proof follows as in the first case with w (or − w ) in place of v$_{2}$ − v$_{1}$ . We let γ ( s ) = φ ( t ) = p and let v$_{1}$ = φ ( t$_{1}$ ) = γ ( s$_{1}$ ) and v$_{2}$ = φ ( t$_{2}$ ) = γ ( s$_{2}$ ) be the neighboring vertices (so t$_{1}$ < t < t$_{2}$ and s$_{1}$ < s < s$_{2}$ ). As above, there exist s ′ $_{1}$, s ′ 2 such that s$_{1}$ ≤ s ′ 1 ≤ s ≤ s ′ 2 ≤ s$_{2}$ , γ ( s ) − γ ( s$_{1}$ ) ∈ T$_{γ}$ ( s ′ $_{1}$) and γ ( s$_{2}$ ) − γ ( s ) ∈ T$_{γ}$ ( s ′ $_{2}$). Note that T$_{φ}$ ( t ) is exactly the set of positive linear combinations of these vectors. By lemma 4.2.13, for every w ∈ T$_{φ}$ ( t ), there is a s ′ ∈ [ s ′ $_{1}$, s ′ $_{2}$] such that w ∈ T$_{γ}$ ( s $^{′}$). As d ( p, γ ( s $^{′}$)) < ϵ $_{3}$, the proof is complete. Definition 4.3.2. We say that Ω is tangentially graph-like and Lipschitz (TGLL) with radius r if Ω is tangentially graph-like with radius r and there is some constant 0 < K < ∞ such that for every p ∈ ∂ Ω , the arc D ( p, r ) ∩ ∂ Ω is the graph of a Lipschitz function (in the same orientation used by the tangentially graph-like definition) and that the Lipschitz constant is at most K . Remark 4.3.3. Note that tangentially graph-like does not imply tangentially graph-like and Lipschitz: taking γ to be a square with side length 5 whose corners are replaced by quarter circles of radius 1 and then considering disks of radius √$_{2}$ centered on γ yields one example. Because γ is arclength parameterized by s , || γ $^{′}$( s ) || = 1 for all s . Since γ is assumed C 1 on its compact domain [0 , L ], γ ′ is uniformly continuous: for any ϵ > 0, there is a δ$_{ϵ}$ such that if | s$_{2}$ − s$_{1}$ | < δ$_{ϵ}$ then || γ $^{′}$( s$_{2}$ ) − γ $^{′}$( s$_{1}$ ) || < ϵ . 97