= Strict disk-void bound for a Poisson line process
{title2=$\mathbb P(\Phi\cap D_r\ne\varnothing)<1-(1+2\pi cr)e^{-2\pi cr}$}
For $c,r>0$, an intersection inside a disk requires at least two lines to meet that disk. The latter count is <Poisson> with <mean> $2\pi cr$, giving the non-strict bound first. It is strict because two lines with nearby normal angles but separated perpendicular distances can both hit the disk while intersecting outside. Choose two positive-measure parameter boxes forcing this configuration; the event of exactly one line in each box and no other disk-hitting line has positive <probability>. This also yields a void <probability> too large to equal that of a <Poisson point process> with the intersection process's <intensity measure>.
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