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Traffic Plans and Distances between Measures

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1955)

Abstract

In this chapter, we consider the irrigation and who goes where problems for the cost functional Eα introduced at the end of Chapter 3. We prove in Section 6.1 that for \( \alpha > 1 - \tfrac{1} {N} \) where N is the dimension of the ambient space, the optimal cost to transport µ+ to μ is finite. More precisely, if μ+ and μ are two nonnegative measures on a domain X with the same total mass M and \( \alpha > 1 - 1/N \), set

$$ E^\alpha \left( {\mu ^ + ,\mu ^ - } \right): = \mathop {\min }\limits_{\chi \in TP\left( {\mu ^ + ,\mu ^ - } \right)} E^\alpha \left( \chi \right). $$
((6.1))

Then (μ+ can be bounded by

$$ E^\alpha \left( {\mu ^ + ,\mu ^ - } \right) \leqslant C_{\alpha ,N} M^\alpha diam\left( X \right). $$

The proof of this property, first proven in [94], follows from the explicit construction of a dyadic tree connecting any probability measure on X to a Dirac mass. If α is under this threshold it may happen that the iofimum is in fact ∞.

Keywords

  • Probability Measure
  • Edge Length
  • Ambient Space
  • Dirac Mass
  • Triangular Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2009 Springer-Verlag Berlin Heidelberg

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(2009). Traffic Plans and Distances between Measures. In: Optimal Transportation Networks. Lecture Notes in Mathematics, vol 1955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69315-4_6

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