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where α = ˆ r/ ˆ h . Notice that the above set up works everywhere except when φ ( x ) = b , in which case we obtain an orthogonal projection for φ$_{π}$ ( x ) along b − a . Choosing coordinates for the tangent spaces of σ and τ$_{x}$ to be { v$_{1}$ , w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ , v$_{2}$ } and { w $^{′}$, w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ } , respectively, we get from Equation (2.15) that Dφ$_{π}$ ( x , a ) is the ( ℓ − 1) × ℓ matrix given as  

## Bounding the Integral of the Jacobian 

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We now present a series of bounds on integrals of the dilation of d -volumes induced by the retraction. Since ℓ = d implies we are already in the d -skeleton and no retraction is needed, we can assume that ℓ > d . We start with a bound on the maximum dilation of d -volumes under the retraction φ . Dφ will denote the tangent map or Jacobian map of φ . 

Definition 2.5.5. Let J$_{d}$φ ( x , a ) be the maximum dilation of d -volumes induced by Dφ ( x , a ) at x . 

We will use the definitions and results on Dφ$_{π}$ ( x , a ) in ℓ -dimension given above. In particular, recall that diameter( σ ) is the diameter of σ , ˆ h = ‖ b − a ‖ and h = ‖ y − a ‖ . 

Lemma 2.5.6. For any center a and any point x ̸ = a in the ℓ -simplex σ with d < ℓ ≤ p = d + k , 

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