If I'm not mistaken, one can argue that $d^2/2$ is strictly convex if $d$ is convex. And with strict convexity, for every $x$ we deduce the uniqueness of $argmin_{a\in A} d^2(x,a)/2$, from which uniqueness of $argmin_{a\in A} d(x,a)$ follows. The strict convexity of quadratic distance $d^2/2$ versus the convexity of $d$ is basically why $L^2$ optimal transport is so regular and $L^1$ optimal transport more difficult.

@PhilHarmsworth Is there a particular critique in the above url link that you find especially persuasive? I am aware of the Crothers/Hooft correspondance, although their exchanges are mutually quite rude. Being skeptical and honest means I'm obligated to thoroughly study BOTH sides of the controversy. What do you think is Crothers' strongest argument?

It's possible that the article by S.J. Crothers' vixra.org/abs/1101.0013 contains an example of static vacuum universe which contains a visible curvature singularity at r=2m, see Section 2, equation (3). Crothers' research is based on earlier observations of L.Abrams, and the distinction between Schwarzschild's original solution, and Hilbert's later solution which is miscredited to Schwarzschild, and which makes the physical error of confusing the quantity R=1+r with the units of ``radius". Crothers is controversial, but his paper vixra.org/abs/1103.0051 is excellent.

Look again at the inequality $-\phi(x)+\psi(y)\leq c(x,y)$ for all $x\in X$, $y\in Y$. If $h(x,y)$ is any function satisfying $-\phi(x)+\psi(y)\leq c(x,y)+h(x,y)$ for all $x,y$ and obtains equality iff $(x,y)\in spt(\pi)$, then the new cost $c':=c+h$ will again have $\pi$ as $c'$-optimal transport (... if i'm not mistaken).

It is strange question, for if the $c$-optimal coupling $\pi$ is observable, then are the dual Kantorovich potentials $-\phi=\psi^c$, $\psi^{cc}=\psi$ also observable? For we have always $-\phi(x)+\psi(y)\leq c(x,y)$ for all $x\in X$, $y\in Y$ with equality iff $x\in \partial^c \psi(y)$ iff $y\in \partial^c \phi(x)$ iff $(x,y)\in spt(\pi)$. Therefore if $\phi, \psi$ are observable, then you have pointwise lower bound for $c(x,y)$, namely $-\phi(x)+\psi(y)$ for arbitrary $x,y$. All the above answers show the problem is underdetermined, with many arbitrary parameters.

@abx yes thank you that is useful observation. For complex dimension 2, i'm basically looking for representation of $GL(2,3)$ into $Sp(\mathbb{Z}^4)$. For $\dim=3$ a representation of the automorphisms of the $A_6^2$ lattice, and for $\dim=4$ want automorphisms of $E_8$ lattice. Will read Conway/Sloane closer.