Figure 2.5: Illustration for integration of Jacobian bound over centers a instead of x ’s. For the case of the triangle shown, there will be 3 sets of 3 regions. In general there will be 3 regions for every face of the simplex. guaranteed to cover B σ as some of B σ may occupy parts of Region 3. Since a ∈ B $_{σ}$, we have ˆ h > r$_{σ}$ , and h ′ ≤ h when a ∈ Region 3, so that Combining the above estimates while integrating over all such Regions 2 and 3 gives us the bound specified in the Lemma. We consider the d -current T , and employ the bounds on the Jacobian of retraction described above to the pushforwards of T and its boundary ∂T on to the simplicial 38 Bounding the pushforwards of the current $^{(}$h ′ h$^{)}$$^{d}$ diameter( σ ) ˆ h ≤ diameter( σ ) r$_{σ}$ . x a Region 3 Region 2 Region 1 φ ( x ) h h ′ ˆ h D( σ )