Regularity of the positional penalization function in inter-sign optimal transport on real measures
We study the Monge--Kantorovich optimal transport problem between two signed measures~$\mu$ and~$\nu$ on convex compact subsets of~$\mathbb{R}^d$, with a positional penalization function~$\lambda(x, y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(\pi^{++}, \pi^{+-}, \pi^{-+}, \pi^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(\mu, \nu)$ is weakly-$*$ compact and non-empty if and only if $\mu^+(X) = \nu^+(Y)$ and~$\mu^-(X) = \nu^-(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$\lambda$ at the intersection of inter-sign supports. The penalization~$\lambda$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge--Amp\`ere equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $\sigma \det(D^2_{yx}\Lambda) e > 0$. The classical Brenier equation is recovered in the limit~$\lambda \to 0$.