Figure 3.3: A sequence of simplicial chains that converges in the flat norm (i.e., P$_{n}$ → T ) need not have convergent simp licial flat norm values (i.e., F$_{K}$$_{n}$ ( P$_{n}$ ) → F ( T ) need not hold). The current T is the segmen t from A to B , the complex K$_{n}$ is the arrangement of 2 n equilateral triangles of appropriate size stretching from A to B and P$_{n}$ is the top chain from A to B on K$_{n}$ . Clearly , F ( T − P$_{n}$ ) → 0 but F$_{K}$$_{n}$ ( P$_{n}$ ) = 2 √ $_{3}$F ( T ) ̸→ F ( T ). A K$_{2}$ B T P$_{2}$ Figure 3.4: Various appro ximations and decompositions used in our results. T P$_{δ}$ X + ∂S X$_{δ}$ + ∂S$_{δ}$ = Optimal flat norm decomp osition Polyhedral approximation X → X$_{δ}$ S S$_{δ}$ Polyhedral approximation flat norm of an integral chain in codimension 1 has an optimal integral current decom- position; by the compactness theorem from geometric measure theory, the limit of these decompositions is also integral. In order to show that an in tegral current T has integral flat norm decomposition, we therefore find suitable simplicial approximations to T and take the limit of their simplicial flat norm decomp ositions to obtain an integral decomposition for T . We must also show that this decomposition achieves the flat norm value for T (that is, express T using integral curren ts in such a way that it remains an optimal flat norm decomposition). This is immediate if our simplicial approximations to T have simplicial flat norm values that converge to the flat norm of T but this is not necessary (see 55