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Gradient Flows pp 133-150 | Cite as

The Optimal Transportation Problem

Part of the Lectures in Mathematics ETH Zürich book series (LM)

Abstract

Let X, Y be separable metric spaces such that any Borel probability measure in X, Y is tight (5.1.9), i.e. Radon spaces, according to Definition 5.1.4, and let c : X × Y → [0,+] be a Borel cost function. Given μ Open image in new window (X), ν Open image in new window (Y) the optimal transport problem, in Monge’s formulation, is given by
$$ \inf \left\{ {\smallint _X c\left( {x,t\left( x \right)} \right)d\mu \left( x \right):t_\# \mu = \nu } \right\}. $$
(6.0.1)
This problem can be ill posed because sometimes there is no transport map t such that t#μ = ν (this happens for instance when μ is a Dirac mass and ν is not a Dirac mass). Kantorovich’s formulation
$$ \min \left\{ {\smallint _{X \times Y} c\left( {x,y} \right)d\gamma \left( {x,y} \right):\gamma \in \Gamma \left( {\mu ,\nu } \right)} \right\} $$
(6.0.2)
circumvents this problem (as μ× ν ∈ Г(μ, ν)). The existence of an optimal transpoplan, when c is l.s.c., is provided by (5.1.15) and by the tightness of Г(μ, ν) (this property is equivalent to the tightness of μ, ν, a property always guaranteed in Radon spaces).

Keywords

Gaussian Measure Borel Probability Measure Optimal Transport Superlinear Growth Kantorovich Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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