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 $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,α)$ in the case $α\in[π/2,π]$ and an upper bound for $p(n,m,α)$ in the case $α\in [0,π/2]$ which tends to $p(n,m,π/2)$ when $α\toπ/2$. In the case $α\in[0,π/2]$ this yields upper bounds for the expected number of spherical caps of radius $α$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ 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.