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Flat Norm Decomposition of Integral Currents
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 -dimensional current
into - and (the boundary of) -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
-currents that are boundaries of -currents in
(following results from a corresponding problem on the total variation
(TV) 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
TV approach.
We develop an analysis framework that extends the result in the simplicial
setting to one for -currents in , provided a suitable
triangulation result holds. In , 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
-currents in .Comment: 17 pages, adds some related work and application
A continuous model of transportation revisited
We review two models of optimal transport, where congestion effects during
the transport can be possibly taken into account. The first model is Beckmann's
one, where the transport activities are modeled by vector fields with given
divergence. The second one is the model by Carlier et al. (SIAM J Control Optim
47: 1330-1350, 2008), which in turn is the continuous reformulation of
Wardrop's model on graphs. We discuss the extensions of these models to their
natural functional analytic setting and show that they are indeed equivalent,
by using Smirnov decomposition theorem for normal 1-currents.Comment: 26 pages. Theorem A.20 of v1 was not correct: we removed it and
replaced it with the counterexample A.18 in v2. We also made some
improvements to the wording and corrected some typo
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