For t representing the set T with integer multiplicity of one, t$_{i}$ ∈ {− 1 , 0 , 1 } with − 1 indicating that the orientations of σ$_{i}$ and T are opposite. But t$_{i}$ can take any integer value in general. Thus, t is the representation of T in the elementary d -chain basis of K . We consider ( d + 1)-chains in K modeling sets S representing rectifiable ( d + 1)-currents with integer multiplicities, and denote them similarly by ∑ n j $_{=1}$s$_{j}$ τ$_{j}$ in the elementary ( d + 1)-chain basis of K consisting of the individual simplices τ$_{j}$ . We denote the chain modeling such a set S using the corresponding vector of weights s ∈ Z $^{n}$. Relationships between the d - and ( d + 1)-chains of K are captured by its ( d + 1)- boundary matrix [ ∂$_{d}$$_{+1}$ ], which is an m × n matrix with entries in {− 1 , 0 , 1 } . If the d -simplex σ$_{i}$ is a face of the ( d + 1)-simplex τ$_{j}$ , then the ( i, j ) entry of [ ∂$_{d}$$_{+1}$ ] is nonzero, otherwise it is zero. This nonzero value is +1 if the orientations of σ$_{i}$ and τ$_{j}$ agree, and is − 1 when their orientations are opposite. The d -chain representing the set T − ∂$_{d}$$_{+1}$S is then given as Notice that x ∈ Z $^{m}$. We define the simplicial flat norm (SFN) of T represented by the d -chain t in the ( d + 1)-dimensional simplicial complex K as Since x and s are chains in a simplicial complex, the masses of the currents they represent (as given in Equation 2.1) are indeed given by the weighted sums of the volumes of the corresponding simplices. The integer restrictions x ∈ Z m and s ∈ Z n are important in this definition as we are studying currents with integer multiplicities. The simplicial flat norm is intuitively the problem of deforming an input chain to another chain of least cost, where cost is determined both by the mass of the resulting chain and the size of the 18 F$_{S}$ ( T ) = min s ∈$_{Z}$ n    m ∑ i =1 V$_{d}$( σ$_{i}$ ) | x$_{i}$ | + n ∑ j =1 V$_{d}$$_{+1}$( τ$_{j}$ ) ∣ $_{∣}$s$_{j}$ ∣ ∣ x = t − [ ∂$_{d}$$_{+1}$ ] s , x ∈ Z m    . (2.4) x = t − [ ∂$_{d}$$_{+1}$ ] s .