## The Annals of Probability

### Coverage processes on spheres and condition numbers for linear programming

#### 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.

#### Article information

Source
Ann. Probab., Volume 38, Number 2 (2010), 570-604.

Dates
First available in Project Euclid: 9 March 2010

https://projecteuclid.org/euclid.aop/1268143527

Digital Object Identifier
doi:10.1214/09-AOP489

Mathematical Reviews number (MathSciNet)
MR2642886

Zentralblatt MATH identifier
1205.60027

#### Citation

Bürgisser, Peter; Cucker, Felipe; Lotz, Martin. Coverage processes on spheres and condition numbers for linear programming. Ann. Probab. 38 (2010), no. 2, 570--604. doi:10.1214/09-AOP489. https://projecteuclid.org/euclid.aop/1268143527

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