content_image/322927.jpg content_image/322925.jpg content_image/322926.jpg content_image/322923.jpg content_image/322924.jpg content_image/322922.jpg
Recall that X + ∂S = T is an optimal flat norm decomposition of T and X$_{δ}$ and S$_{δ}$ are polyhedral approximations to X and S on our simplicial complex K$_{δ}$ . Using the decompositions in Equations (3.16) and (3.18), we can write: 

<img src='content_image/322927.jpg'>

Now apply Theorem 3.3.6 with ϵ = 1 to the currents U δ i and W δ j for all i ∈ { 0 , 1 } and j ∈ { 0 , 1 , 2 } to obtain U$_{i,δ}$ and W$_{j,δ}$ on the simplicial complex K$_{δ}$ with 

<img src='content_image/322925.jpg'>

<img src='content_image/322926.jpg'>

Applying Equations (3.17) to (3.19), we obtain the following from Equation (3.21): 

<img src='content_image/322923.jpg'>

<img src='content_image/322924.jpg'>

We apply the linearity result of Theorem 3.3.6 to Equation (3.20) along with the fact that P$_{δ}$ , X$_{δ}$ , and ∂S$_{δ}$ are fixed by projection to the d -skeleton of K$_{δ}$ to yield 

<img src='content_image/322922.jpg'>

which, as all quantities are chains on K$_{δ}$ , is a candidate simplicial flat norm decom- position of P$_{δ}$ . Using this observation, the triangle inequality, and Equations (3.17) 

