Random triangles in planar regions

摘要

In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point O.These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point O. The formulae provide another way to approach the Sylvester’s four-point problem. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: 2/27 +20 ln 2/81 (Halász and Kleitman in Stud Appl Math 53:225– 237, 1974; Prékopa in Period Math Hung 2:259–282, 1972).We compute this probability in the case of a regular polygon and its center of mass for the point O. Other families of regions are studied. For the family of Lima?ons r = a + cos t, a > 1, and O the origin of the polar coordinates, the probability is 1 4 ? 12a2(4a2+1)/(2a2+1)3π2 .