Gangbo, W. 1994. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mechanics Anal., 128:381--399.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and existence for a class of mass transportation problems and economic applications G. Carlier 19th December 2000 Abstract We establish duality, existence and uniqueness results for a class of mass transportations problems. We extend a technique of W. Gangbo [9] using the Euler Equation of the dual problem. This is done by introducing the h Fenchel Transform and using its basic properties. The cost functions we consider satisfy a generalization of the so called Spence Mirrlees condition which is well known by economists in dimension 1. We therefore end ....

....R esum e Nous etablissons dans cet article des r esultats de dualit e, d existence et d unicit e pour une classe de probl emes de transport optimal de masse. La nouveaut e r eside ici dans l emploi de la transform ee de Fenchel h convexe qui permet d utiliser un argument de W. Gangbo [9] consistant a exploiter l equation d Euler du probl eme dual. Les couts de transport que nous consid erons satisfont une condition g en eralisant la condition de Spence Mirrlees bien connue des economistes en dimension 1. Nous terminons ainsi cet article par une application de notre r esultat a ....

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W. Gangbo. An elementary proof of the polar factorization of vectorvalued functions, Arch. Rational Mech. Anal., vol. 128 (1994).

.... = cc (with OE = c ) from which we shall construct a map minimizing Monge s cost. Compactness of the manifold facilitates a direct RJMc Polar Factorization of Maps on Manifolds May 27, 1999 4 proof of the existence of ( OE) and the duality relation (5) following an approach of Gangbo [17] also developed in Caffarelli [8] 9] and Gangbo and McCann [18] We give this proof after some preliminaries on infimal convolutions with c. A key ingredient is the change of variables formula for pushed forward measures s # = Z M h d(s #) Z M h(s(x) d(x) 8) holds for all Borel s : M ....

....and hence t : M Gamma M are Borel maps defined a.e. in view of Lemma 4. One way to prove that the map t(x) exp x [ Gammar (x) pushes forward to is to integrate each continuous function h 2 C(M) against the measures and t # and show the two integrals coincide. We do this following Gangbo [17], Gangbo and McCann [18] and Caffarelli [9] For x; y 2 M and jfflj 1 define perturbations OE ffl (y) c (y) fflh(y) and ffl : OE ffl ) c by ffl (x) inf y2M c(x; y) Gamma OE(y) Gamma fflh(y) 21) Fix a point x where 0 is differentiable. Continuity and compactness of M ....

W. Gangbo. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mech. Anal. 128 (1994) 381--399.

.... 1] fi fi fi OE and are lower semicontinuous OE(x) y) x Delta y for all x; y 2 Omega o : With help of this dual problem, Brenier was able to show that there is a unique optimal transfer plan and that it is actually in the class of one to one transfer plans I( 0 ; 1 ) see also [8]) Theorem 1 (Brenier) Let 0 ; 1 2 K be given. 16 F. OTTO i) There exists a unique p 2 P ( 0 ; 1 ) satisfying (3.1) It is given by a Phi 2 I( 0 ; 1 ) in the following way Z i(x; y) p(dx dy) Z Omega 0 (x) i(x; Phi(x) dx : ii) There exists (OE; 2 I satisfying (3.2) ....

W. Gangbo, An elementary proof of the polar factorization of vector--valued functions, Arch. Rational Mech. Anal. 128, 1994, pp. 381 -- 399.

....can be said about this optimal transference plan and the Wasserstein metric: Brenier [2] has shown uniqueness of the optimal transference plan in the case of p = 2 and proved that its support is concentrated on the graph of the gradient of a convex function. Caffarelli [3] and Gangbo and McCann [7, 8, 9] have extended Brenier s result to more general cost functions including the p th power of the euclidean distance considered here; we will need their result (Proposition A.3) The case of p = 1 is qualitatively different; it has been investigated by Evans and Gangbo [6] It has been known ....

W. Gangbo, An elementary proof of the polar factorization of vector-- valued functions, Arch. Rational Mech. Anal., 128 (1994), 381 -- 399.

....space dimensions: Brenier [9] has shown uniqueness of the optimal transference plan and proved that its support is concentrated on the graph of the gradient of a convex function (the reader will get a flavor of this result in the proof of Lemma A. 1 b) Caffarelli [10] and Gangbo and McCann [16, 17] have extended Brenier s result to more general cost functions than the squared Euclidean distance. It has been known to the probabilists since a long time that the Wasserstein metric is actually a metric (we will prove the triangle inequality in Lemma A.1 c) and that it metrizes the topology ....

W. Gangbo, An elementary proof of the polar factorization of vector-- valued functions, Arch. Rational Mech. Anal., 128 (1994), 381 -- 399.

....and holds in full generality. Yet, in order to help understanding how the Wasserstein distance is behaving in the limit 0, and why the preceding result is natural, we present another line of reasoning, which we explicit only in the Euclidean case. Closely related considerations appear in [14]. Let r be an optimal gradient of convex function transporting onto (see [23] for instance) This means Z jx r (x)j 2 d (x) W ( 2 : In particular, 1 2 Z jx r (x)j 2 d (x) 1 2 W ( 2 2 H( j ) 2 kfk 2 L 2 (d ) 64) ....

Gangbo, W. An elementary proof of the polar factorization of vector-valued functions. Arch. Rat. Mech. Anal. 128 (1994), 381-399.

....each (x; y) 2 corresponds to a hyperplane which touches but does not cross the graph of . The convex function arises as a Lagrange multiplier to the constraints and ; its existence was originally deduced from the duality theory of Kantorovich [13] For alternative approaches consult Gangbo [10] or McCann [16] As Brenier also realized, when is absolutely continuous with respect to Lebesgue, so is differentiable on a set domr ae R d 1 of full measure, the optimizer fl in (2) will be unique: it full mass lies on the graph f(x; r (x) j x 2 domr g of the gradient of . This means ....

W. Gangbo. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mech. Anal. 128, 381--399 (1994).

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Gangbo, W. 1994. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mechanics Anal., 128:381--399.

No context found.

W. Gangbo, "An elementary proof of the polar factorization of vector-valued functions, " Arch. Rational Mechanics Anal. Vol. 128, pp. 381--399, 1994.

No context found.

W. Gangbo, An elementary proof of the polar factorization of vector-- valued functions, Arch. Rational Mech. Anal. 128 (1994), 381--399.

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