Abstract
Let μ be a positive Boral measure on ℝN which we assume without loss of generality to have total mass 1. If \( E^\alpha \left( {\delta _S ,\mu } \right) < \infty \) for some α ∈ [0,1], then we say that μ is irrigable with respect to α. In that case, notice that μ is also β-irrigable for β>α. This observation proves the existence of a critical exponent α associated with μ and defined as the smallest exponent such that μ is α-irrigable. The aim of the chapter is to link this exponent to more classical dimensions associated with μ such as the Hausdorff and Minkowski dimensions of the support of μ. Inequalities between these dimensions and an “irrigability dimension” will be established. A striking result is that when μ is Ahlfors regular, all considered dimensions are equal. To illustrate the results, let us take the case where μ is the Lebesgue measure of a ball in dimension N. We already know in that case that μ is irrigable for every \( \alpha > 1 - \tfrac{1} {N} \). What happens if α is critical? Corollary 10.16 gives the answer: if any probability measure μ with a bounded supports is α-irrigable, then α>1/N. Thus the N-dimensional Lebesgue measure is not 1−1/N irrigable. Here the presentation and results follow closely Devillanova’s PhD [28] and Devillanova-Solimini [78].
Keywords
- Probability Measure
- Borel Measure
- Compactness Theorem
- Positive Borel Measure
- Disjoint Ball
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). Irrigability and Dimension. In: Optimal Transportation Networks. Lecture Notes in Mathematics, vol 1955. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69315-4_10
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DOI: https://doi.org/10.1007/978-3-540-69315-4_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69314-7
Online ISBN: 978-3-540-69315-4
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