supported on the ( d − 1) -skeleton (i.e., a simplicial d -chain) such that T − P = Q + ∂R and there exists a constant ϑ$_{K}$ (depending only on simplicial regularity in K ) such that the following controls on mass hold: where Δ is the diameter of the largest simplex in K . The regularity constant ϑ$_{K}$ is given by where for each l -simplex σ , perimeter( σ ) is the ( l − 1) -volume of ∂σ and B$_{σ}$ is the l -volume of a ball with radius inradius( σ ) / 2 in R $^{l}$. Proof highlights. The simplicial current P is obtained by pushing T and its boundary to the d − and ( d − 1)-dimension skeletons of K respectively. This pushing is done one dimension at a time; that is, T is pushed from the p -skeleton (i.e., the full complex K ) to the ( p − 1)-skeleton, then to the ( p − 2) and so on until the d -skeleton. Pushing the current from the ℓ -skeleton to the ( ℓ − 1)-skeleton is done by picking a projection center in each ℓ -simplex σ and projecting the current in σ outwards to ∂σ via straight-line projection. A crucial step in the proof is to find a projection center that bounds the expansion of T and ∂T . In particular, this is done by proving that over all possible centers, the average expansion is bounded and then showing that individual centers exist with 60 ϑ$_{K}$ = sup σ ∈ K diameter( σ ) perimeter( σ ) B$_{σ}$ + 2 sup σ ∈ K diameter( σ ) inradius( σ ) (3.8) M( P ) ≤ (4 ϑ$_{K}$ ) p − $^{d}$M( T ) + Δ(4 ϑ$_{K}$ ) p − d $^{+1}$M( ∂T ) (3.3) M( ∂P ) ≤ (4 ϑ$_{K}$ ) p − d $^{+1}$M( ∂T ) (3.4) M( Q ) ≤ Δ(4 ϑ$_{K}$ ) p − $^{d}$(1 + 4 ϑ$_{K}$ ) M( ∂T ) (3.5) M( R ) ≤ Δ(4 ϑ$_{K}$ ) p − $^{d}$M( T ) (3.6) F ( T, P ) ≤ Δ(4 ϑ$_{K}$ ) p − $^{d}$(M( T ) + (1 + 4 ϑ$_{K}$ ) M( ∂T )) (3.7)