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Covering random points in a unit disk

Part of: Geometric probability and stochastic geometry Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  01 July 2016

Jennie C. Hansen ,
Eric Schmutz  and
Li Sheng
Jennie C. Hansen*
Affiliation:
Herriot-Watt University
Eric Schmutz*
Affiliation:
Drexel University
Li Sheng*
Affiliation:
Drexel University
*
∗ Postal address: Actuarial Mathematics and Statistics Department and the Maxwell Institute for Mathematical Sciences, Herriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK.
∗∗ Postal address: Mathematics Department, Drexel University, Philadelphia, PA 19104, USA.
∗∗ Postal address: Mathematics Department, Drexel University, Philadelphia, PA 19104, USA.
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Abstract

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Let D be the punctured unit disk. It is easy to see that no pair x, y in D can cover D in the sense that D cannot be contained in the union of the unit disks centred at x and y. With this fact in mind, let Vn = {X1, X2, …, Xn}, where X1, X2, … are random points sampled independently from a uniform distribution on D. We prove that, with asymptotic probability 1, there exist two points in Vn that cover all of Vn.

Keywords

Dominating setrandom geometric graphunit ball graph

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 68R10: Graph theory (including graph drawing)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 22 - 30
Copyright
Copyright © Applied Probability Trust 2008 

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