Remark 2.5.11. We did not explicitly discuss the case of 0 -dimensional currents. In this case, the bounds on mass expansion are all equal to one. Sullivan studied the deformation of integral currents on to the skeleton of a cell complex, which is composed of compact convex sets. He presented a deformation theorem for deforming integral currents on to the boundary of a cell complex [42, Theorem 4.5]. For ease of comparison, we use our notation to restate the bounds given by Sullivan for deforming a d -current T to a polyhedral current P in the boundary of a cell complex in R $^{q}$. Recall that in our simplicial deformation theorem (Theorem 2.5.1), the simplicial complex considered has dimension p and is embedded in R q for q ≥ p . Furthermore, κ$_{1}$ , κ$_{2}$ , Δ , and ϑ$_{K}$ are simplicial regularity constants. We also note that even though Sullivan stated his results for full-dimensional complexes and the standard flat norm, it is straightforward to extend them to lower dimensional complexes and the flat norm 43 2.5.2 Comparison of Bounds of Approximation