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My question is similar to this one but for rectangles instead of lines.

Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed random points inside the rectangle, and why?

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The answer, given in (Burgstaller and Pillichshammer 2009; Mathai et al. 1999), is

$$ \frac1{15} \left( \frac{L_w^3}{L_h^2}+\frac{L_h^3}{L_w^2}+d \left( 3-\frac{L_w^2}{L_h^2}-\frac{L_h^2}{L_w^2} \right) +\frac52 \left( \frac{L_h^2}{L_w}\log\frac{L_w+d}{L_h}+\frac{L_w^2}{L_h}\log\frac{L_h+d}{L_w} \right) \right)\;, $$

where $d=\sqrt{L_w^2+L_h^2}$.

REFERENCES:

  • Burgstaller, B. and Pillichshammer, F., "The average distance between two points", Bulletin of the Australian Mathematical Society, 80(3), pp.353-359, 2009.
  • Mathai, A.M., Moschopoulos, P., Pederzoli, G., "Random points associated with Rectangles", Rendiconti del Circolo Matematico di Palermo, II, XLVIII (1999).
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    $\begingroup$ It seems to be a simple problem. But it's not. Good find. (+1) $\endgroup$
    – JACKY Li
    Oct 7 '12 at 14:18
  • $\begingroup$ The famous book "Integral geometry and geometric probability" of Santalo is available on line at (projecteuclid.org/euclid.bams/1183539854). $\endgroup$
    – Jean Marie
    Oct 27 '17 at 10:55

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