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

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.