On a globally hyperbolic spacetime, any weak Kantorovich potential for the squared time-separation cost is locally semiconvex almost everywhere, implying optimal couplings are induced by transport maps.
Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’.Class
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Optimal transport and regularity of weak Kantorovich potentials on a globally hyperbolic spacetime
On a globally hyperbolic spacetime, any weak Kantorovich potential for the squared time-separation cost is locally semiconvex almost everywhere, implying optimal couplings are induced by transport maps.