Fine properties of monotone maps arising in optimal transport for non-quadratic costs

Abstract

The cost functions considered are c(x,y)=h(x−y)c\left(x,y)=h\left(x-y), where h∈C2(Rn)h\in {C}^{2}\left({{\mathbb{R}}}^{n}) is homogeneous of degree p≥2p\ge 2 with a positive definite Hessian in the unit sphere. We study multivalued monotone maps with respect to that cost and establish that they are single-valued almost everywhere. Further consequences are then deduced.

Keywords

• optimal transportation
• monotone maps
• multivalued maps
• rectifiability
• monge problem
• 49q22
• 47h05