OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...
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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