content_image/315419.jpg content_image/315420.jpg content_image/315433.jpg content_image/315422.jpg content_image/315434.jpg content_image/315423.jpg content_image/315424.jpg content_image/315421.jpg
and 

<img src='content_image/315419.jpg'>

<img src='content_image/315420.jpg'>

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 

<img src='content_image/315433.jpg'>

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 

and the following controls on mass M hold: 

where ϑ$_{K}$ = κ$_{1}$ + κ$_{2}$ . 

<img src='content_image/315422.jpg'>

<img src='content_image/315434.jpg'>

<img src='content_image/315423.jpg'>

<img src='content_image/315424.jpg'>

<img src='content_image/315421.jpg'>

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 

