OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Concentration case. The following factsin the concentration case hold:• If µ 0 and µ 1 are convex combinations(also countable) of Dirac masses, thenthey can be connected by a path γ(t) offinite minimal cost J .• If r > 1 − 1/N then every pair of probabilitiesµ 0 and µ 1 can be connected by apath γ(t) of finite minimal cost J .• The bound above is sharp. Indeed, if r ≤1 − 1/N there are measures that cannotbe connected by a finite cost path (forinstance a Dirac mass and the Lebesguemeasure).57
Example. (Y-shape versus V-shape). Wewant to connect (concentration case r < 1fixed) a Dirac mass to two Dirac masses (ofweight 1/2 each) as in figure below, l and hare fixed. The value of the functional J isgiven byJ (γ) = x + 2 1−r√ (l − x) 2 + h 2 .Then the minimum is achieved forhx = l − √41−r− 1 .When r = 1/2 we have a Y-shape if l > hand a V-shape if l ≤ h.•h•xl•A Y-shaped path for r = 1/2.58
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OPTIMIZATION PROBLEMS INMASS TRANSP
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The Monge problem is thenmin{ ∫ d
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Then the class of admissible transp
Page 7 and 8: Monge-Kantorovich problem:{ ∫ }mi
Page 9 and 10: Shape optimization problemsThere is
Page 12 and 13: The shape optimization problem abov
Page 14 and 15: To overcome some of these difficult
Page 16 and 17: In the mass optimization problem th
Page 18 and 19: More precisely, we consider the geo
Page 20 and 21: produces an optimal µ which has a
Page 22 and 23: Variational integrals w.r.t. a meas
Page 24 and 25: Here are some cases where the optim
Page 26 and 27: -2 -1.5 -1 -0.5 0 0.5 1 1.5 21.510.
Page 28 and 29: Problem 1 - Optimal NetworksWe cons
Page 30 and 31: •Large town policy: several ticke
Page 32 and 33: It is interesting to study the opti
Page 34 and 35: The function B can be seen as a con
Page 36 and 37: Here is another case, with a single
Page 38 and 39: • there is a transportation cost
Page 40 and 41: We have then the optimization probl
Page 42 and 43: and so the value of the cost F (a).
Page 44 and 45: is a delicate matter. Indeed, maxim
Page 46 and 47: Some Numerical ComputationsHere are
Page 48 and 49: Optimal sets of length 0.5 and 1 in
Page 50 and 51: Now we give a model that takes into
Page 52 and 53: We also give a model where the oppo
Page 54 and 55: Theorem Assume:• J is lower semic
Page 56 and 57: ∫∫ ∫J(µ) = f(µ a )dm+ f ∞
Page 60 and 61: Diffusion case. The following facts
Page 62 and 63: Theorem Let (X, d, d ′ ) be a met
Page 64 and 65: • if q < 1 + 1/N for every µ 0 a
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OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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