OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

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Example Take the measuresf + = δ A and f − = 1 2 δ B + 1 2 δ C ;it is clear that no map T transports f + intof −so the Monge formulation above is inthis case meaningless.Example Take the measures in R 2 , still singularbut nonatomicf + = H 1 ⌊A and f − = 1 2 H1 ⌊B + 1 2 H1 ⌊Cwhere A, B, C are the segments below.CAB3
Then the class of admissible transport mapsis nonempty but the minimum in the Mongeproblem is not attained. Indeed, if the distancebetween the lines is L and the heightis H it can be seen that the infimum of theMonge cost is HL, while every transportmap T has a cost strictly greater than HL.Example (book shifting) Consider in R themeasuresf + = 1 [0,a] L 1 , f − = 1 [b,a+b] L 1 .Then the two mapsT 1 (x) = b + x,T 2 (x) = a + b − xare both optimal; the map T 1 corresponds toa translation, while the map T 2 correspondsto a reflection. Indeed there are infinitelymany optimal transport maps.4
Page 1 and 2: OPTIMIZATION PROBLEMS INMASS TRANSP
Page 3: The Monge problem is thenmin{ ∫ d
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 58 and 59:
Concentration case. The following 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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