Optimal Transport with Defective Cost Functions with Applications to the Lens Refractor Problem

TL;DR

This paper introduces defective cost functions in optimal transport, explores their properties on various geometries, and applies the theory to the lens refractor problem, including regularity conditions and curvature analysis.

Contribution

It defines defective cost functions, extends their properties to different geometries, and applies the regularity theory to the lens refractor problem.

Findings

Defective cost functions map to points along geodesics.

Cost-sectional curvature can be computed and verified for various functions.

Regularity theory can be developed satisfying MTW conditions for these functions.

Abstract

We define and discuss the properties of a class of cost functions on the sphere which we term defective cost functions. We then discuss how to extend these definitions and some properties to cost functions defined on Euclidean space and on surfaces embedded in Euclidean space. Some important properties of defective cost functions are that they result in Optimal Transport mappings which map to points along geodesics, have a nonzero mixed Hessian term, among other important properties. We also compute the cost-sectional curvature for a broad class of cost functions, to verify and some known examples of cost functions and easily prove positive cost-sectional curvature for some new cost functions. Finally, we discuss how we can construct a regularity theory for defective cost functions by satisfying the Ma-Trudinger-Wang (MTW) conditions on an appropriately defined domain. As we develop the regularity theory of defective cost functions, we discuss how the results apply to a particular instance of the far-field lens refractor problem.