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Mathematics
Mean distance between two points in a domain
N. G. Aharonyan Chair of Probability Theory and Mathematical Statistics YSU, Armenia
Abstract:
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.
Keywords:
chord length distribution function, mean distance, convex domain geometry. Received: 18.06.2012
Accepted: 20.07.2012
Language: English