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  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 , Bounding the Integral of the Jacobian Dφ$_{π}$ ( x , a ) =      α 0 0 ... 0 0 0 1 0 ... 0 0 . . . . . . 0 . . . 1 0$^{}$      . (2.16) J$_{d}$φ ( x , a ) ≤ $^{(}$ˆ h h$^{)}$$^{d}$ diameter( σ ) ˆ h . 34