(Equation 2.5) indeed exists. The function is lower bounded by zero, as it is the sum of nonnegative entries (we have λ ≥ 0 ). Notice that F λ $_{S}$( T ) = min$_{S}$ f $^{λ}$( T, S ) . Further, we only consider integral s defined on the finite simplicial complex K , and hence there are only a finite number of values for this function. Hence its minimum indeed exists, which defines the multiscale simplicial flat norm of t . On the other hand, the proof of existence of minimum in the original definition of flat norm for rectifiable currents employs the Hahn–Banach theorem [19, pg. 367]. We illustrate the optimal decompositions to compute the multiscale simplicial flat norm for two different scales ( λ = 1 and λ ≪ 1) in Figure 2.2. Notice that the input set T , shown in blue, is not a closed loop here. It is a subcomplex of the simplicial complex triangulating Ω. The underlying set Ω need not be embedded in R 2 – it could be sitting in R 3 or any higher dimension. We do not show the orientations of individual simplices and chains so as not to clutter the figure. We could take each triangle to be oriented counterclockwise (CCW), with T oriented CCW as well, and each edge oriented arbitrarily. When scale λ = 1, we get the default SFN of T , where the S chosen (shown in light pink) is such that the resulting optimal T − ∂S (indicated by the thin curve in dark green) is devoid of all the “kinks”, but is similar to T in overall form. This removal of the tightest “kinks” is a discrete analogue of how the λ in the flat norm relates to the curvature in the continuous case. For λ ≪ 1, the second term in the definition (Equation 2.5) contributes much less to the multiscale simplicial flat norm. As such, the optimal T − ∂S consists of a short chain of two edges (shown in light green), which closes the original T curve to form a loop. S in this case includes the triangles in the former choice of S , and all other triangles enclosed by the original curve T and the resulting 20 f $^{λ}$( T, S ) = m ∑ i =1 w$_{i}$ | x$_{i}$ | + λ ( n ∑ j =1 v$_{j}$ ∣ $_{∣}$s$_{j}$ ∣ $_{∣}$) with x = t − [ ∂$_{d}$$_{+1}$ ] s (2.6)