2.5 Simplicial Deformation Theorem When can we use the multiscale simplicial flat norm as a discrete surrogate for the traditional flat norm? That is, if we wish to solve a flat norm problem (for which there are no practical algorithms in general), can we discretize the problem and find a problem close enough to the original one which we can solve? The deformation theorem [19, Sections 4.2.7–9] is one of the fundamental results of geometric measure theory, and more particularly of the theory of currents. It approxi- mates an integral current by deforming it onto a cubical grid of appropriate mesh size. On the other hand, we have been studying currents or sets in the setting of simplicial complexes, rather than on square grids. Our proof is a substantial modification of the classical proof of the deformation theorem. We found the presentation of the latter proof by Krantz and Parks [29, Section 7.7] especially helpful. Our proof mimics their proof when possible. The gist of this theorem is the assertion that we may approximate a current with a simplicial current. Recall that V$_{d}$( σ ) denotes the d -dimensional volume of a d -simplex σ . The perime- ter of σ is the set of all its ( d − 1)-dimensional faces, denoted as perimeter( σ ) = {∪ $_{j}$τ$_{j}$ | τ$_{j}$ ∈ σ, dim( τ$_{j}$ ) = d − 1 } . We will also refer to the ( d − 1)-dimensional volume of perimeter( σ ) as the perimeter of σ , but denote it as P( σ ) = ∑ τ$_{j}$ ∈ perimeter( σ $_{)}$V$_{d}$$_{−}$$_{1}$ ( τ$_{j}$ ). We let diameter( σ ) be the diameter of σ , which is the largest Euclidean distance between any two points in σ . diameter( σ ) P( σ ) V$_{d}$( B $_{σ}$) ≤ κ$_{1}$ < ∞ , Theorem 2.5.1 ( Simplicial Deformation Theorem). Let K be a p -dimensional simplicial complex embedded in R $^{q}$, with p = d + k for k ≥ 1 and q ≥ p . Suppose that for every simplex σ ∈ K 29