## Coverage processes on spheres and condition numbers for linear programming.(English)Zbl 1205.60027

Let $$p(n,m,\alpha)$$ be the probability that $$n$$ spherical caps of angular radius $$\alpha$$ do not cover the whole sphere $$S^m$$. The authors give an exact formula for this probability in the case $$\alpha\in[\pi/2,\pi]$$ and an upper bound in the case $$\alpha\in[0,\pi/2]$$, which tends to the exact value as $$\alpha\to\pi/2$$. The obtained upper bound yields upper bounds for the expected number of caps that are needed to cover the whole sphere.
Then the authors study the condition number $$\mathcal{C}(A)$$ of the linear programming feasibility problem concerning the existence of a nonzero $$x\in\mathbb{R}^{m+1}$$ such that $$Ax\leq 0$$ componentwise, where a $$n\times(m+1)$$ matrix $$A$$ is randomly chosen according to the standard normal distribution. The relation to spherical caps is explained by the fact that $$\mathcal{C}(A)$$ equals $$1/|\cos\rho(A)|$$, where $$\rho(A)$$ is the radius of a largest excluding cap for $$A=(a_1,\dots,a_n)$$, i.e. the complement to the smallest cap that contains the points $$a_1,\dots,a_n$$. The authors exactly determine the distribution of $$\mathcal{C}(A)$$ conditioned on $$A$$ being feasible and provide an upper bound on the distribution function in the infeasible case. From these results they show that $\mathbf{E} \ln\mathcal{C}(A)\leq 2 \ln(m+1)+3.31\,,\quad n>m\,,$ which is the sharpest known bound for this expectation.

### MSC:

 60D05 Geometric probability and stochastic geometry 52A22 Random convex sets and integral geometry (aspects of convex geometry) 90C05 Linear programming
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### References:

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