particular, we must avoid integrating over a close to the perimeter of σ . Hence we integrate over as big a region as we can while still avoiding a neighborhood of the perimeter. As in the statement of the main Theorem 2.5.1, let ˆ B σ be the largest ball inscribed in σ , B σ be the ball with half the radius and same center as ˆ B $_{σ}$, and r$_{σ}$ be the radius of B $_{σ}$. Lemma 2.5.8. For any point x in the ℓ -simplex σ with d < ℓ ≤ p = d + k , Proof. Similar to the subsimplices of σ considered in the Proof of Lemma 2.5.7, let σ$_{j}$ now denote the ℓ -simplex formed by x and τ$_{j}$ ∈ perimeter( σ ). In order to derive an upper bound, we integrate instead over regions that are by construction bigger than these subsimplices of σ . Denoting the simplex σ$_{j}$ as Region 1, we define Regions 2 and 3 as follows. We refer the reader to Figure 2.5 for an illustration of this construction. Let σ ′ j be the reflection of σ$_{j}$ through x , and similarly, let τ ′ j be the reflection through x of τ$_{j}$ . We define σ ′ j as Region 2. Notice that unlike Region 1, Region 2 need not be contained fully in σ . As defined in Section 2.5.1, let z be the unit vector normal to the ( ℓ − 1)-hyperplane containing τ$_{j}$ pointing into σ . We define Region 3 as the ℓ -dimensional set τ ′ j + [0 , diameter( σ )] z , as illustrated in Figure 2.5. Note that the union of all Region 2’s and Region 3’s cover σ . By an argument almost identical to that above, we have the following upper bound on the integrand in question. Integrating the second of these two terms over Region 2 and summing the integral over all such Regions 2 for all faces τ$_{j}$ , we get the upper bound of diameter( σ ) P( σ ). Here we use the same arguments as the ones employed in Lemma 2.5.7. Region 2 alone is not 37 $^{(}$h ′ h ) d ∥ $_{∥}$φ ( x , a ) − a ∥ ˆ h ≤ $^{(}$h ′ h ) d diameter( σ ) ˆ h . ∫ B$_{σ}$ J$_{d}$φ ( x , a ) d L $^{ℓ}$( a ) ≤ diameter( σ ) P( σ ) + V$_{ℓ}$( B $_{σ}$) diameter( σ ) r$_{σ}$ .