The geometry of optimal transportation
type: Item
instance of: scholarly article
title: The geometry of optimal transportation
zbMATH Open document ID: 0887.49017
DOI: 10.1007/BF02392620
author: Wilfrid Gangbo
author: Robert J. McCann
published in: Acta Mathematica
publication date: 1997-11-05T00:00:00Z
review text: This paper treats a classical (1781) problem of Monge whose essence is: Let \((X,\mu)\) and \((Y,\nu)\) be (probability) measure spaces, where \(\mu\) represents the distribution of the production of some commodity at various ``localities'' \(x\in X\) and \(\nu\) of the consumption of that commodity at localities \(y\in Y\). There is a cost function \(c(x,y)\) on \(X\times Y\) which represents the cost of transporting the commodity from \(x\) to \(y\). A measure-preserving transformation \(s:X\to Y\) represents a ``distribution scheme'' for the commodity, resulting in a total transport cost of \(C_1(s)= \int c(x,s(x))d\mu(x)\). \textit{Monge's problem} is to find, where possible, the (or an) \(s\) which gives the minimum \(M_1\) of \(C_1(s)\) over all such choices of \(s\). \textit{Kantorovich's problem}, simpler than Monge's one, is to find, where possible, a measure \(\gamma\) on \(X\times Y\) which has \(\mu\) and \(\nu\) as marginals, and which gives the minimum \(M_2\) of the transport cost \(C_2(\gamma)= \int c(x,y)d\gamma(x, y)\) over all such choices of \(\gamma\). It is easy to see that \(M_2\leq M_1\), but there are cases where \(M_1= M_2\) and where \(s\) can be accessed through \(\gamma\). Sometimes \(s\) is even uniquely determined a.e. For their main (positive) results, the authors restrict themselves to \(X, Y\subseteq\mathbb{R}^d\), \(d= 1,2,3,\dots\), and to two classes of cost functions: (i) \(c(x,y)= h(x-y)\), where \(h\) is strictly convex (relatively easy); (ii) \(c(x,y)= l(|x-y|)\), where \(|\cdot|\) denotes Euclidean distance on \(\mathbb{R}^d\), and where \(l\geq 0\) is strictly concave (harder but more realistic in the economic context). They also give numerous examples where the (unique) map \(s\) can be explicitly constructed, and their treatment is largely self-contained.
Mathematics Subject Classification ID: 49J99
Mathematics Subject Classification ID: 91B60
Mathematics Subject Classification ID: 90B06
zbMATH DE Number: 1083669
zbMATH Keywords: cost functions
zbMATH Keywords: minimum-cost transformations
zbMATH Keywords: Monge's problem
zbMATH Keywords: Kantorovich's problem
zbMATH Keywords: measure-preserving transformation
Wikidata QID: Q63508067
MaRDI profile type: Publication
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cites work: Polar factorization and monotone rearrangement of vector‐valued functions
cites work: The Regularity of Mappings with a Convex Potential
cites work: Notes on the Wasserstein metric in Hilbert spaces
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cites work: Optimal coupling of multivariate distributions and stochastic processes
cites work: A characterization for the solution of the Monge--Kantorovich mass transference problem
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full work available at URL: https://doi.org/10.1007/bf02392620
OpenAlex ID: W2096413780
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description: scientific article; zbMATH DE number 1083669