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Johan Philip derives the PDF for the distance between 2 random points in a unit square: http://www.math.kth.se/~johanph/habc.pdf

What is the PDF for the minimum separation of 3 random points in a unit square? (where the maximum minimum separation is known to be sqrt(6)-sqrt(2): http://en.wikipedia.org/wiki/Circle_packing_in_a_square

Similar question for k points where 3 <= k <= 9 (for which the maximum minimum separation is known precisely). Similar question for unit circle.

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I know you seek the PDF in an explicit form, but here is the distribution empirically, for 100,000 trials, of the minimum separation among three points within a unit square:
 Histogram min 3 pts
The mean minimum distance is 0.305 and the median is 0.287.

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    $\begingroup$ Unsurprisingly, looks Poisson $\endgroup$ – Igor Rivin Oct 29 '13 at 3:24
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    $\begingroup$ Thanks. We've simulated the PDF and CDF for 10^7 points for k from 3 through 6. It looks like a Weibull or a Beta (suitably normalized to [0,1]). The CDF of a Weibull seems to match the data exactly for 10^6 simulations. Also a PP-plot looks like a straight line. However, a Kolmogorov-Smirnoff test rejects the null hypothesis that it's a Weibull (as estimated from maximum likelihood) at the 0.01 level. $\endgroup$ – user87456 Oct 29 '13 at 17:50

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