Motivated by the financial problem of building financial portfolios which outperform the market,Pal and Wong considered optimal transport on the probability simplex \triangle^n where the cost function is induced by the free energy. We study the regularity of this problemand find that the associated MTW tensor is non-negative definite and in fact constant on \triangle^n \times \triangle^n. We further find^n equipped with probability measures \mu and \nu, respectively. We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y }\to  of the form c(x,y)= \Psi(x-y) for some convex function \Psi. For this problem, we find an associated K\"ahler manifold whose orthogonal holomorphic bisectional curvature is proportional to the so-called MTW tensor, which was introduced by Ma, Trudinger, and Wang and plays an essential role in the regularity theory of optimal transport. We also show } that relative c-convexity corresponds to the standard notion of convexity in the probability simplex. Hence, we are able to use standard results in optimal transport to establish regularity for the optimal transport maps considered by Pal and Wong. We also provide several new examples of costs satisfying the MTW(0) condition  .
Let X and Y be domains of \mathbb{R^n equipped with probability measures \mu and \nu, respectively. We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y }\to \mathbb{R of the form c(x,y)= \Psi(x-y) for some convex function \Psi. For this problem, we find an associated K\"ahler manifold whose orthogonal holomorphic bisectional curvature is proportional to the so-called MTW tensor, which was introduced by Ma, Trudinger, and Wang and plays an essential role in the regularity theory of optimal transport. We also show } that relative c-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to \mathbb{C pseudo-arbitrages, or investment strategies which outperform the market .
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Motivated by the financial problem of building financial portfolios which outperform the market,Pal and Wong considered optimal transport on the probability simplex \triangle^n where the cost function is induced by the free energy. We study the regularity of this problemand find that the associated MTW tensor is non-negative definite and in fact constant on \triangle^n \times \triangle^n. We further find
Let X and Y be domains of \mathbb{R
A
\mathbb{R
R
corresponds to the standard notion of convexity in the probability simplex. Hence, we are able to use standard results in optimal transport to establish regularity for the optimal transport maps considered by Pal and Wong. We also provide several new examples of costs satisfying the MTW(0) condition
geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to \mathbb{C
A
pseudo-arbitrages
A
, or investment strategies which outperform the market