diameter( σ ) r$_{σ}$ ≤ κ$_{2}$ < ∞ , diameter( σ ) ≤ Δ T − P = Q + ∂R, M( P ) ≤ (4 ϑ$_{K}$ ) $^{k}$M( T ) + Δ (4 ϑ$_{K}$ ) k $^{+1}$M( ∂T ) , (2.10) M( R ) ≤ Δ (4 ϑ$_{K}$ ) $^{k}$M( T ) , and (2.12) M( Q ) ≤ Δ (4 ϑ$_{K}$ ) $^{k}$(1 + 4 ϑ$_{K}$ ) M( ∂T ) , (2.13) Remark 2.5.2. It is immediate that the flat norm distance between T and P can be made arbitrarily small by subdividing the simplicial complex to reduce Δ while preserving the regularity of the refinement as measured by κ$_{1}$ and κ$_{2}$ . Remark 2.5.3. Note that this theorem combines the unscaled and scaled versions of the 30 and the following controls on mass M hold: Then there exists a simplicial d -current P supported in the d -skeleton of K whose bound- ary ∂P is supported in the ( d − 1) -skeleton of K such that hold, where ˆ B σ is the largest ball inscribed in σ , B σ is the ball with half the radius and same center as ˆ B $_{σ}$, and r$_{σ}$ is the radius of B $_{σ}$. Let T be a d -dimensional current in R q such that the support of T is a subset of the underlying space of K . Suppose that T satisfies and where ϑ$_{K}$ = κ$_{1}$ + κ$_{2}$ . M( T ) + M( ∂T ) < ∞ . M( ∂P ) ≤ (4 ϑ$_{K}$ ) k $^{+1}$M( ∂T ) , (2.11)