Random segments in the Euclidean plane (Q1197538)

Suppose \(K\) is a convex \(n\)-gon in the Euclidean plane, \(\mathcal L\) an unoriented segment of length \(l\), midpoint at \((x,y)\), which makes an angle \(\theta\) with respect to some fixed direction. \(\mathcal L\) lies at random in the sense of the kinematic density \(dx\land dy\land d\theta\). Assume \(\mathcal L\) cannot meet any two consecutive sides of \(K\). The author computes by classical methods the measure of \(\{{\mathcal L}: {\mathcal L}\subset K\}\) and uses this to prove that if \(\mathcal L\) lies at random on a lattice with \(K\) as its fundamental cell, then \(\mathcal L\) meets a cell boundary with probability \(\{4l{\mathcal L}-l^ 2\sum^ n_{i=1}(1+(\pi-\alpha_ i)\text{ctn }\alpha_ i\}/S\). Here \(\mathcal L\), \(S\) denote the perimeter and area of \(K\), respectively, \(\alpha_ 1,\dots,\alpha_ n\) its interior angles. He generalizes this to the case in which \(l\) is random with a specified probability density and with given first and second moments. Finally he uses some of these results to calculate the variance of the random variable \(S_ m\) equal to the area of \(K_ 0,K_ 1,\dots,K_ m\), where \(K_ 0\) is a circle contained in \(K\) and \(K_ 1,\dots,K_ m\) are random congruent copies of \(K\).