stylistic:style_transfer, information:highlighting, linguistic:paraphrasing, structural:scholar_edition
Optimal transportation on the real line

For 1 ≤ p < ∞ {\displaystyle 1\leq p<\infty }, let P p (R) {\displaystyle {\mathcal {P}}_{p}(\mathbb {R})} denote the collection of probability measures on R {\displaystyle \mathbb {R} } that have finite p {\displaystyle p} -th moment. Let μ, ν ∈ P p (R) {\displaystyle \mu,\nu \in {\mathcal {P}}_{p}(\mathbb {R})} and let c (x, y) = h (x − y) {\displaystyle c(x,y)=h(x-y)}, where h: R → [ 0, ∞) {\displaystyle h:\mathbb {R} \rightarrow [0,\infty)} is a convex function.

The proof of this solution appears in Rachev & Rüschendorf (1998).