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bounded expansion. We call out this particular step because we modify it to obtain the next theorem. 

When projecting onto the skeleton of each simplex σ , we have [24, Lemma 5.9] 

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where B σ is the set of possible centers in σ and ϑ$_{σ}$ is a regularity constant for σ related to ϑ$_{K}$ by ϑ$_{K}$ = sup$_{σ}$$_{∈}$$_{K}$ ϑ$_{σ}$ . This shows that in each projection step the mass of T expands by a factor of at most ϑ$_{K}$ averaged over all possible choices of centers. As the average expansion over all centers is ϑ$_{K}$ , we observe that at most 1 4 of the possible centers can expand the mass of T by a factor of 4 ϑ$_{K}$ or more. Similarly, at most 1 4 of the centers can expand ∂T by a factor of 4 ϑ$_{K}$ or more. Therefore, at least 1 2 of the possible centers bound the expansion of both T and ∂T by at most a factor of 4 ϑ$_{K}$ . Choosing a center from this set for each simplex yields the bounds required in the theorem. 

The following theorem allows normal (or integral) currents to be approximated by polyhedral chains which are simplicial chains not necessarily contained in an a priori complex. Note in particular that the mass bounds can be made arbitrarily tight by choice of ϵ in contrast with the larger bounds of the deformation theorems. 

Theorem 3.2.5 (Polyhedral approximation of currents [19], 4.2.21, 4.2.24) . If ρ > 0 and T is a normal m -current in R n supported in the interior of a compact subset K of R $^{n}$, then there exists a polyhedral chain P with 

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