Kantorovich's optimal transportation problem

\begin{equation} \tau_c(\mu,\nu)=\min\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}c(x,y)d\pi(x,y) \end{equation} where $\Pi(\mu,\nu) = \{\pi\in P(X\times Y); \pi(A\times Y) = \mu(A), \pi(X\times B) = \nu(B) \}$

It is a well-studied topic when cost function $c$ is non-negative. But if $c$ is actually a non-positive function, I could not find any literature researching it. Namely, The problem is equivalent to the maximum cost transport \begin{align} M_c(\mu,\nu)&:=\max\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}\vert c(x,y)\vert d\pi(x,y)\\ &=-\min\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}-\vert c(x,y)\vert d\pi(x,y)\\ &=-\min\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}c(x,y) d\pi(x,y) \end{align}

This problem seems quite different from the optimal transport problem in the sense that it is well-defined and it can yield non-trivial solutions. For example, consider the maximum cost transport in the interval $[0,1]$, with cost function $c(x,y)=\vert x-y\vert$

$X=Y=[0,1]$, and $d\mu(x)=dx$, then obviously, $M_1(\mu,\mu)$ is not $0$, and \begin{equation} M_1(\mu,\mu)\le\int_{[0,1]\times [0,1]}1d\pi(x,y)=1 \end{equation}

Actually I can prove that $M_1(\mu,\mu)=\frac12$. I think that $M_1(\mu,\mu)=0$ if and only if $\mu$ is point measure. It seems that $M_c$ could describe how "dispersive" or "concentrated" a measure is.

Moreover, some of the useful theorems in classical optimal transport theory still hold for the maximal cost transport, for example, I can prove that the duality formulation of Kantorovich's problem still holds for maximal cost transport. Namely,

Theorem (Duality formulation of maximal cost transport) Let $X$ and $Y$ be Polish spaces, let $\mu\in P(X), \nu\in P(Y)$, and let $c:X\times Y\rightarrow \mathbb R_+\cup \{\infty\} $ be a continuous cost function.Then \begin{equation} \sup\limits_{\Pi(\mu, \nu)}\int_{X\times Y}c(x,y)d\pi(x,y)=\inf\limits_{\Phi_c}\int_X\phi d\mu+\int_Y \psi d\nu \end{equation} where $$\Pi(\mu,\nu) = \{\pi\in P(X\times Y); \pi(A\times Y) = \mu(A), \pi(X\times B) = \nu(B) \}$$ and $$\Phi_c:=\{(\phi,\psi)\in L^1(d\mu)\times L^1(d\nu):\phi(x)+\psi(y)\ge c(x,y)\}$$

Any literature about the maximum cost transport would be appreciated.

4more comments