Uniqueness conditions for C1 regularizers in divergence-regularized optimal transport

Establish whether uniqueness up to an additive constant of the optimal potentials holds for divergence-regularized optimal transport when the convex conjugate ψ is only C1, by rigorously proving conditions analogous to the connected-support assumption that guarantees uniqueness in the quadratic regularization case. This is needed to extend the central limit theorems derived under C2 assumptions to broader regularizers.

Background

The paper’s central limit theorems rely on uniqueness of the optimal potentials, which the authors prove under ψ∈C2 via Lemma 4.44. They note that their results would remain valid for C1 regularizers provided a suitable uniqueness condition is available.

For quadratic regularization, an analogous uniqueness result is known when at least one measure has connected support (Nutz, 2025). The authors explicitly speculate that similar conditions might ensure uniqueness for other C1 regularizers but emphasize that this requires a rigorous proof.