Mean distance between two points in a domain

N. G. Aharonyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2012), pp. 3-8

Résumé

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

Détail

Keywords: chord length distribution function, mean distance, convex domain geometry.