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If i was to give an $n×n$ grid with each grid point having probability $p$ of being selected, would it be difficult to calculate distributions of various measures regarding the convex hull of all selected points? I.e the distribution of the number of extreme points or total area covered (assuming this is defined appropriately). I've seen a lot of stuff on convex hulls in a continuous setting but not much in a discrete setting, if anyone could mention any interesting things they may know, that would be much appreciated.

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    $\begingroup$ Scott, “On Convex Lattice Polygons,” proves that if $i$ is the number of interior lattice points and $b$ the number of boundary lattice points, then $b < 2 i + 7$. $\endgroup$ Jun 22, 2014 at 11:25
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    $\begingroup$ A good starting point is arxiv.org/pdf/0912.0631v1.pdf as they give a good historical review. In a different direction, articles by Barany will give you also a lot of information, for example his ICM talk arxiv.org/abs/math/0304462 $\endgroup$ Jun 22, 2014 at 11:25
  • $\begingroup$ The journal version of that Imre Bárány paper is: "Random points and lattice points in convex bodies." Bulletin of the American Mathematical Society 45.3 (2008): 339-365. $\endgroup$ Jun 22, 2014 at 13:20
  • $\begingroup$ @ Joseph O'Rourke and Ofer zeitouni, thank you very much for those useful links $\endgroup$ Jun 22, 2014 at 13:40

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