In weak optimal transport, convex (or increasing convex) dual potentials are exactly characterized by the cost being decreasing in (increasing) convex order, with attainment under mild regularity.
Backhoff-Veraguas, M
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Weak Optimal Transport: When is the Dual Potential Convex?
In weak optimal transport, convex (or increasing convex) dual potentials are exactly characterized by the cost being decreasing in (increasing) convex order, with attainment under mild regularity.