Thus we can take C$_{θ}$ = 48 π cot( θ/ 2) 2 + 4 cot( θ/ 2). Our result relies on the ability to localize irregularities via subdivision, focusing on localization rather than removal because the latter is not possible. For example, any subdivision of a 2-complex with a very small input angle will have an angle that is at least as small. With that in mind, we require that subdivisions be possible which push the irregularities into the corners. That is, the irregularity should be bounded by a constant (independent of the complex) away from the original complex skeleton and a complex- dependent constant (reflecting the necessity of some bad simplices) near the skeleton. Conjecture 3.3.4 formalizes this requirement and Theorem 3.3.5 notes some cases where it holds. We present our main theorem in such a way that proving Conjecture 3.3.4 more generally will automatically extend our results. Conjecture 3.3.4. For any p -dimensional simplicial complex K in$_{R}$ q and ϵ > 0 , it is possible to subdivide K so that all simplices are of bounded “badness” (with bound inde- pendent of K or ϵ ) except possibly for simplices in a region of p -dimensional volume less than ϵ near the ( p − 1) -skeleton; even these simplices have bounded badness (dependent on K but not ϵ ). More precisely, there exists a subdivision M$_{ϵ}$ of K and a subcomplex M ′ ϵ of M$_{ϵ}$ (with simplicial regularity constants ϑ$_{M}$$_{ϵ}$ and ϑ$_{M}$ ′ $_{ϵ}$) such that: In particular, α$_{K}$ does not depend on ϵ and β does not depend on K or ϵ . The simplicial regularity constants are defined as in Equation (3.8) . Theorem 3.3.5. Conjecture 3.3.4 holds for: 67 1. M$_{ϵ}$ \ M ′ ϵ ⊆ { x ∈ R q | ‖ x − y ‖ < ϵ for some y in the ( p − 1) -skeleton of K } , 2. ϑ$_{M}$$_{ϵ}$ ≤ α$_{K}$ for some constant α$_{K}$ , 3. ϑ$_{M}$ ′ ϵ ≤ β for some fixed constant β .