Allocation Maps with General Cost Functions

Introduction: In the theory of allocations one is interested to transfer a domain Ω₁ into a domain Ω₂ (one can think on sending k points X ₁, …,X _k in a one to one fashion to k points Y ₁,Y _k ), minimizing certain cost function C(X,Y). (See for instance Rachev ([R]). For C ( X , Y ) = 1 2 | X − Y | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203744369/d9858e60-9246-4876-bb34-38e3d28ecc5f/content/eq204.tif"/> , this problem has interest in several fields, and a complete existence theory was given by Brenier ([B]) and a regularity theory was developed by the author ([C1], [C2]).