MEAN DISTANCE BETWEEN TWO POINTS IN A DOMAIN

Abstract

 Let D be a bounded convex domain in the Euclidean plane and we choose uniformly and independently two points in D. How large is the mean distance m(D) between these two points? Up to now, there were known explicit expressions for m(D) only in three cases, when D is a disc, an equilateral triangle and a rectangle. In the present paper a formula for calculation of mean distance m(D) by means of the chord length density function of D is obtained. This formula allows to find m(D) for those domains D, for which the chord length distribution is known. In particular, using this formula, we derive explicit forms of m(D) for a disc, a regular triangle, a rectangle, a regular hexagon and a rhombus.