content_image/307211.jpg content_image/307210.jpg
and (3.22), we have 

<img src='content_image/307211.jpg'>

The claim follows from taking the limit as δ ↓ 0. 

Claim 3.3.7.3. For each δ > 0 , there exist integral simplicial chains Y$_{δ}$ and R$_{δ}$ on K$_{δ}$ such that P$_{δ}$ = Y$_{δ}$ + ∂R$_{δ}$ is an optimal simplicial flat norm decomposition (i.e., F$_{K}$$_{δ}$ ( P$_{δ}$ ) = M( Y$_{δ}$ ) + M( R$_{δ}$ ) ). 

Proof of claim. This follows from Theorem 3.2.3. 

Claim 3.3.7.4. There exists c > 0 such that for all δ ≤ 1 , the currents Y$_{δ}$ , ∂Y$_{δ}$ , R$_{δ}$ , and ∂R$_{δ}$ all have mass at most c . 

Proof of claim. Using the fact that P$_{δ}$ = Y$_{δ}$ + ∂R$_{δ}$ is an optimal simplicial flat norm decomposition and facts from Equation (3.17), we observe that 

<img src='content_image/307210.jpg'>

