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

Abstract The cost functions considered are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>h</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> c\left(x,y)=h\left(x-y) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> h\in {C}^{2}\left({{\mathbb{R}}}^{n}) is homogeneous of degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> p\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.