PAIR OF LINES AND MAXIMAL PROBABILITY

Abstract

In this paper we consider two independent and identically distributed lines, which intersect a planar convex domain D. We evaluate the probability P_D, for the lines to intersect inside D. Translation invariant measures generating random lines is obtained, under which P_D achieves its maximum for a disc and a rectangle. It is also shown that for every p from the interval [0,1/2] and for every square there are measures generating random lines such that P_D = p.