Chapter 3 Flat norm decomposition of integral currents 1 3.1 Introduction In geometric measure theory, currents represent a generalization of oriented surfaces with multiplicities. Currents were developed in the context of Plateau’s problem and have also found application in isoperimetric problems and soap bubble conjectures [32]. Given a d -dimensional current T , we can consider decompositions T = X + ∂S where X is a d -dimensional current and S is a ( d +1)-dimensional current. Over all such decompositions, the minimum total mass (volume) of the two pieces (i.e., M( X )+M( S )) is the flat norm F ( T ). More recently, the L $^{1}$TV functional (introduced in the form most relevant to us by Chan and Esedo¯ glu [9]) was shown to be related to the flat norm [34]. This connection suggested the flat norm with scale (yielding the objective M( X ) + λ M( S ) for any fixed scale λ ) and a geometric interpretation for the optimal decompositions: varying λ controls the scale of features isolated in the decomposition. One natural question: must currents in a particular regularity class (in this paper, integral currents) have an optimal flat norm decomposition in the same class? The L $^{1}$TV connection shows this is true for boundaries of codimension 1 (i.e., boundaries of ( d + 1)-currents in R d $^{+1}$) since the L $^{1}$TV functional applied to binary (or step function) input is known to have binary (step function) minimizers [9]. This may be taken one step further in the discretized problem where the boundary requirement can be dropped [24]. $^{1}$Based on [25] 51