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The projection operator is clearly linear and commutes with the boundary operator as a consequence of properties [19, 4.1.6] of the differential forms to which currents are dual. 

Corollary 3.3.2. The bounds in Theorem 3.2.4 can all be tightened by replacing 4 ϑ$_{K}$ with (2 + ϵ ) ϑ$_{K}$ . 

Proof. Simply taken m = 1 and n = 0 in Theorem 3.3.1. 

For a 2-complex K , the minimum angle over all triangles in the complex is eas- ier to work with and can be used as a proxy for our simplicial regularity constant as Lemma 3.3.3 indicates. 

Lemma 3.3.3. A lower bound on the minimum angle of all triangles in a 2-complex implies an upper bound on the simplicial regularity constant. That is, given a 2-complex K with minimum angle at least θ , we have ϑ$_{K}$ ≤ C$_{θ}$ for some constant C$_{θ}$ . 

Proof. The simplicial regularity constant ϑ$_{K}$ used for Theorems 3.2.4 and 3.3.1 in the case of triangles is given by 

<img src='content_image/302257.jpg'>

We observe that bounding diameter( σ ) / inradius( σ ) and perimeter( σ ) / inradius( σ ) for all triangles σ ∈ K yields a bound for ϑ$_{K}$ . Suppose σ has side lengths a ≥ b ≥ c and angle γ opposite c . Using the law of cotangents, we obtain 

<img src='content_image/302258.jpg'>

The bound for perimeter( σ ) / inradius( σ ) follows easily from this: 

<img src='content_image/302259.jpg'>

