On the Kähler Geometry of Certain Optimal Transport Problems
Abstract
Let $X$ and $Y$ be domains of $\mathbb{R}^n$ equipped with respective probability measures $\mu$ and $ \nu$. We consider the problem of optimal transport from $\mu$ to $\nu$ with respect to a cost function $c: X \times Y \to \mathbb{R}$. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly \textit{relatively $c$convex} with respect to each other and cost function has nonnegative \textit{MTW tensor}. For cost functions of the form $c(x,y)= \Psi(xy)$ for some convex function $\Psi$, we find an associated Kähler manifold whose orthogonal antibisectional curvature is proportional to the MTW tensor. 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 pseudoRiemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete Kähler surface with nonnegative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to $\mathbb{C}^2$. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \textit{pseudoarbitrages}, or investment strategies which outperform the market.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1812.00032
 Bibcode:
 2018arXiv181200032K
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Differential Geometry;
 Quantitative Finance  Mathematical Finance;
 49Q20;
 53C55;
 46N30;
 46N10;
 35J96
 EPrint:
 30 pages. In the previous versions, there was a switched index in the curvature formulas. We have fixed the issue in this version