  Multi-marginal optimal transport plans are concentrated on c-splitting sets.
It is known that, similar to the two-marginal case, c-splitting sets are
c-cyclically monotone. Within a suitable framework, the converse implication
was very recently established by Griessler. However, for an arbitrary cost c,
given a multi-marginal c-cyclically monotone set, the question whether there
exists an analogous explicit construction to the one from the two-marginal case
of c-splitting potentials is still open. When the margins are one-dimensional
and the cost belongs to a certain class, Carlier proved that the two-marginal
projections of a c-splitting set are monotone. For arbitrary products of sets
equipped with cost functions which are sums of two-marginal costs, we show that
the two-marginal monotonicity condition is a sufficient condition which does
give rise to an explicit construction of c-splitting potentials. Our condition
is, in principle, easier to verify than the one of multi-marginal c-cyclic
monotonicity. Various examples illustrate our results. We show that, in
general, our condition is sufficient; however, it is not necessary. On the
other hand, we conclude that when the margins are one-dimensional equipped with
classical cost functions, our condition is a characterization of c-splitting
sets and extends classical convex analysis.
