Removing the bounded-target hypothesis in the qualitative uniqueness theorem

Determine the classes of cost functions c: R^d × R^d → R for which the hypothesis that the support of the target measure μ is bounded can be removed from Theorem 1 (which asserts uniqueness up to an additive constant of Kantorovich potentials when the support of the source measure ρ is rectifiably connected, c ∈ C^1, and suitable integrability bounds hold).

Background

Theorem 1 establishes qualitative uniqueness of dual Kantorovich potentials under the assumptions that supp ρ is rectifiably connected, supp μ is bounded, and c ∈ C^1 with an integrable bound |c(x,y)| ≤ a(x)+b(y). The proof uses boundedness of supp μ to ensure local Lipschitz regularity of ρ-side potentials and that the projection of optimal plans’ supports onto the x-variable covers supp ρ.

The authors ask whether this boundedness requirement on supp μ can be relaxed for broader families of costs (e.g., beyond settings where existing techniques control mass sent to infinity), while preserving uniqueness up to constants.