content_image/325211.jpg content_image/325210.jpg
<img src='content_image/325211.jpg'>

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 

<img src='content_image/325210.jpg'>

Combining the above estimates while integrating over all such Regions 2 and 3 gives us the bound specified in the Lemma. 

## Bounding the pushforwards of the current 

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 

