Coverage Processes on Spheres and Condition Numbers for Linear Programming

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 S do not cover the whole sphere S. 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 α → π/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 S. Secondly, we study the condition number C (A) of the linear programming feasibility problem ∃x ∈ R Ax ≤ 0, x 6= 0 where A ∈ R is randomly chosen according to the standard normal distribution. We exactly determine the distribution of 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 E(lnC (A)) ≤ 2 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.