content_image/301864.jpg content_image/301872.jpg
Proof. We apply properties of norms to obtain 

<img src='content_image/301864.jpg'>

Letting n → ∞ yields the linearity result. Now let X$_{n}$ and S$_{n}$ be m - and ( m +1)-currents such that X$_{n}$ + ∂S$_{n}$ is a flat norm decomposition of T$_{n}$ − T for n = 1 , 2 , . . . , observing that 

The boundary result follows in the limit. 

<img src='content_image/301872.jpg'>

In the case of the simplicial flat norm, an input integral chain is guaranteed an integral chain decomposition whenever the simplicial complex is totally unimodular [24]. This occurs when the complex is free of relative torsion which is the case for any ( d + 1)-complex in R d +1 or when triangulating a compact, orientable ( d + 1)-dimensional manifold. 

Theorem 3.2.3 (Simplicial flat norm integral decomposition [24]) . If K is a simplicial ( d + 1) -complex embedded in R d $^{+1}$, then for any integral d -chain P on K , the optimal simplicial flat norm value for P is attained by an integral decomposition. 

We state the simplicial deformation theorem and sketch a portion of its proof. We will later modify it to obtain a multiple current deformation theorem that preserves linearity (Theorem 3.3.1). 

Theorem 3.2.4 (Simplicial deformation theorem [24]) . Suppose K is a p -dimensional simplicial complex in$_{R}$ q and T is a normal d -current supported on the underlying space of K . There exists a simplicial d -current P supported on the d -skeleton of K with boundary 

