# Kantorovich Dual Problem (for c(x,y) = d(x,y)^2 where d is a metric)

Kantorovich Dual Problem (for ${\displaystyle c(x,y)=d(x,y)^{2}}$ where ${\displaystyle d}$ is a metric).

The case where the cost is given by ${\displaystyle c(x,y)={\frac {1}{2}}|x-y|^{2}}$ is treated as a special case due to its relationship to c-concavity and convexity. [1]

## Relationship to c-concavity and convexity

### Proposition

Given a function ${\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} \cup \{+\infty \}}$, define ${\displaystyle u_{f}:\mathbb {R} ^{d}\to \mathbb {R} \cup \{+\infty \}}$ by ${\displaystyle u_{f}(x)={\frac {1}{2}}|x|^{2}-f(x)}$. Then ${\displaystyle u_{f^{c}}=(u_{f})^{*}}$. Then a function ${\displaystyle \zeta }$ is c-concave if and only if ${\displaystyle x\mapsto {\frac {1}{2}}|x|^{2}-\zeta (x)}$ is convex and lower semicontinuous. [1]

## Theorem

### Theorem

Let ${\displaystyle \mu ,\nu }$ be probabilities over ${\displaystyle \mathbb {R} ^{d}}$ and ${\displaystyle c(x,y)={\frac {1}{2}}|x-y|^{2}}$. Suppose ${\displaystyle \int |x|^{2}dx,\int |y|^{2}dy<+\infty }$, which implies min(KP) ${\displaystyle <+\infty }$ and suppose that ${\displaystyle \mu }$ gives no mass to ${\displaystyle (d-1)}$ surfaces of class ${\displaystyle C^{2}}$. Then there exists a unique optimal transport map ${\displaystyle T}$ from ${\displaystyle \mu }$ to ${\displaystyle \nu }$, and it is of the form ${\displaystyle T=\nabla u}$ for a convex function ${\displaystyle u}$. [1]