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Distribution of Distances between Elements in a Compact Set

Author

Listed:
  • Solal Lellouche

    (UFR Physique, Université Paris-Diderot, 75013 Paris, France)

  • Marc Souris

    (UMR Unité des Virus Emergents (UVE Aix-Marseille Univ-IRD 190-Inserm 1207-IHU Méditerranée Infection), 13005 Marseille, France)

Abstract

In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.

Suggested Citation

  • Solal Lellouche & Marc Souris, 2019. "Distribution of Distances between Elements in a Compact Set," Stats, MDPI, vol. 3(1), pages 1-15, December.
    • Handle: RePEc:gam:jstats:v:3:y:2019:i:1:p:1-15:d:302199
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    References listed on IDEAS

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    1. Joel E. Cohen & Daniel Courgeau, 2017. "Modeling distances between humans using Taylor’s law and geometric probability," Mathematical Population Studies, Taylor & Francis Journals, vol. 24(4), pages 197-218, October.
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