content_image/314737.jpg content_image/314736.jpg
• q ≥ p = 1 

• p = q = 2 

Proof. The p = 1 case is trivial as all 1-simplices have the same regularity so we have ϑ$_{K}$ = 8 and can take M$_{ϵ}$ = M ′ ϵ = K . 

For the p = q = 2 case, we proceed in two steps. First we will superimpose a square grid on K (orienting it to bound the minimum angle created between its edges and those of K ), creating a cell complex which is a refinement of K . Next we use Shewchuk’s Terminator algorithm [38] to further refine the cell complex back into a simplicial complex with bounds on the minimum angle and, crucially, restrictions on where these small angles can be so that we can obtain regularity bounds. 

By superimposing a fine enough square grid, we can force the small angles (whether already present in the complex or newly created) to occur only in a small measure subset of the complex. Pick δ > 0 small enough that the set 

<img src='content_image/314737.jpg'>

has measure less than ϵ . Let G be a finite square grid in R 2 whose cells each have diameter δ such that G covers the underlying space of K in any rotation. Note that there are only two directions present in G so if we bound all possible angles created between these directions and the edges of K , we can bound the minimum new angle created by superimposing G . 

Let w ∈ R 2 be a fixed unit vector and define 

<img src='content_image/314736.jpg'>

Further let E θ = { ψ ∈ [0 , 2 π ) | | φ − ψ | < θ for some φ ∈ E } . This is the set of angles to 

