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March 2010 Coverage processes on spheres and condition numbers for linear programming
Peter Bürgisser, Felipe Cucker, Martin Lotz
Ann. Probab. 38(2): 570-604 (March 2010). DOI: 10.1214/09-AOP489

Abstract

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in Sm do not cover the whole sphere Sm. We give an exact formula for p(n, m, α) in the case α∈[π/2, π] and an upper bound for p(n, m, α) in the case α∈[0, π/2] which tends to p(n, m, π/2) when απ/2. In the case α∈[0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover Sm.

Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem ∃x∈ℝm+1Ax≤0, x≠0 where A∈ℝn×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all n>m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

Citation

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Peter Bürgisser. Felipe Cucker. Martin Lotz. "Coverage processes on spheres and condition numbers for linear programming." Ann. Probab. 38 (2) 570 - 604, March 2010. https://doi.org/10.1214/09-AOP489

Information

Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1205.60027
MathSciNet: MR2642886
Digital Object Identifier: 10.1214/09-AOP489

Subjects:
Primary: 52A22, 60D05, 90C05

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 2 • March 2010
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