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Given a closed set on the plane and two probability distributions
on the real line, when are there random variables with the given
distributions whose joint distribution is supported by the given set?

We consider both discrete and continuous distributions; in the
latter case, the problem is equivalent to asking which sets
in the unit square can support a permuton, i.e., a Borel distribution
with uniform marginals.

Joint work with Chris Coscia and Martin Tassy.