Area Enclosed by the Convex Hull of a Set of Random Points
source link: https://mathoverflow.net/questions/93099/area-enclosed-by-the-convex-hull-of-a-set-of-random-points
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3 Answers
Update: By a result of Buchta (Zufallspolygone in konvexen Vielecken, Crelle, 1984; available on digizeitschriften.de) there is a general formula for this expected value, it is 1−83(n+1)(n+1∑k=11k(1−12k)−1(n+1)2n+1) yielding (starting with n=3): 11/144, 11/72, 79/360, 199/720, and so on.
The paper contains in fact a more general result, where the problem is solved for any convex m-gon; not just the square.
For asymtotics see other answer(s).
Old version (highly incomplete and wrong guess)
For n=3 the expected value is 11/144 and for n=4 it is 11/72.
This information is taken from a somewhat recent paper (2004) by Johan Philip where the respective distribution functions are studied in detail. I did not see any mention of exact values for other small values of n there (the asymptocic result given already is mentioned though), so they might be unknown.
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