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Random Geometric Graphs$
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Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

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Random Geometric Graphs

Mathew Penrose (Contributor Webpage)

Oxford University Press

This chapter contains some well-known deterministic geometric results for use in subsequent chapters. The Lebesgue density theorem is stated along with some of its consequences concerned with finding disjoint balls of high or low volume. Bounds are given for the covering and packing numbers of a compact (d-1)-submanifold in d-space. The Brunn-Minkowski inequality is stated and some of its consequences are derived, namely an isoperimetric inequality in the square and an isodiametric inequality. A lower bound is given for the volume of the difference between a shifted compact set and the original set.

Keywords:   geometric results, Lebesgue density theorem, Brunn-Minkowski inequality, isoperimetric inequality, isodiametric inequality, submanifold

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