Families of costs with zero and nonnegative MTW tensor in optimal transport
This work provides new families of cost functions for optimal transport on Euclidean and curved spaces, enabling explicit computations and applications in geometry and analysis.
The authors computed the MTW tensor for optimal transport costs of the form c(x,y)=u(x^t y) on R^n, reducing the vanishing condition to a linear ODE and identifying solutions like Lambert and inverse hyperbolic functions, with explicit optimal maps. They extended this to hyperbolic space and spheres, deriving new strictly regular cost families including power functions and analyzing examples like sinh-type costs.
We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on $\mathbb{R}^n$ with a cost function of form $\mathsf{c}(x, y) = \mathsf{u}(x^{\mathfrak{t}}y)$, where $\mathsf{u}$ is a scalar function with inverse $\mathsf{s}$, $x^{\ft}y$ is a nondegenerate bilinear pairing of vectors $x, y$ belonging to an open subset of $\mathbb{R}^n$. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form $\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$ with constant coefficients $P$ and $S$. The resulting inverse functions include {\it Lambert} and {\it generalized inverse hyperbolic\slash trigonometric} functions. The square Euclidean metric and $\log$-type costs are equivalent to instances of these solutions. The optimal map for the family is also explicit. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of $\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of {\it power function costs}. We analyze the $\sinh$-type hyperbolic cost, providing examples of $\mathsf{c}$-convex functions and divergence.