Currents represent generalized surfaces studied in geometric measure theory.
They range from relatively tame integral currents representing oriented compact
manifolds with boundary and integer multiplicities, to arbitrary elements of
the dual space of differential forms. The flat norm provides a natural distance
in the space of currents, and works by decomposing a d-dimensional current
into d- and (the boundary of) (d+1)-dimensional pieces in an optimal way.
Given an integral current, can we expect its flat norm decomposition to be
integral as well? This is not known in general, except in the case of
d-currents that are boundaries of (d+1)-currents in Rd+1
(following results from a corresponding problem on the L1 total variation
(L1TV) of functionals). On the other hand, for a discretized flat norm on a
finite simplicial complex, the analogous statement holds even when the inputs
are not boundaries. This simplicial version relies on the total unimodularity
of the boundary matrix of the simplicial complex -- a result distinct from the
L1TV approach.
We develop an analysis framework that extends the result in the simplicial
setting to one for d-currents in Rd+1, provided a suitable
triangulation result holds. In R2, we use a triangulation result of
Shewchuk (bounding both the size and location of small angles), and apply the
framework to show that the discrete result implies the continuous result for
1-currents in R2.Comment: 17 pages, adds some related work and application