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( ℓ − 1)-dimensional volume of the perimeter, respectively, and Int( σ ) its interior. 

Lemma 2.5.7. For any fixed center a in the ℓ -simplex σ with d < ℓ ≤ p = d + k , 

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Proof. Consider the ( ℓ − 1)-dimensional faces τ$_{j}$ of σ , with perimeter( σ ) = {∪ $_{j}$τ$_{j}$ | τ$_{j}$ ∈ σ, dim( τ$_{j}$ ) = ℓ − 1 } . Let σ$_{j}$ denote the ℓ -simplex generated by a and τ$_{j}$ . Then 

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Let τ$_{j}$ ( h ) denote the ( ℓ − 1)-simplex formed by the intersection of σ$_{j}$ and the ( ℓ − 1)- hyperplane parallel to τ$_{j}$ at a distance h from a . Thus, τ$_{j}$ ( ˆ h ) is τ itself. We observe that our bound on J$_{d}$φ ( x , a ) is constant in τ$_{j}$ ( h ) for any h . The ( ℓ − 1)-dimensional volume of τ$_{j}$ ( h ) is given by 

Using the bound on J$_{d}$φ ( x , a ) from Lemma 2.5.6, and noting that diameter( σ$_{j}$ ) ≤ diameter( σ ) ∀ j , we get 

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Summing this quantity over all τ$_{j}$ ∈ perimeter( σ ) and replacing ℓ − d ≥ 1 with 1 gives the overall bound. 

We now bound the integral of J$_{d}$φ ( x , a ) over centers a with a fixed x that we are retracting onto perimeter( σ ). Examination of the corresponding proof for the original deformation theorem [29, Section 7.7] shows that symmetry of the cubical mesh plays a very special role, which cannot be duplicated in the case of simplicial complex. In 

