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If there are $8$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of a subset in which the points form the vertices of a convex polygon? Thanks!

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By the way, "size" only means cardinality here. – user0o Jan 17 at 20:14
Related: en.wikipedia.org/wiki/Happy_ending_problem – Sam Hopkins Jan 17 at 20:22
@Berlusconi: it does not need to 16. An answer for any number less than or equal to $8$ would be highly appreciated. It may not be trivial to solve for a large number, but one can probably have some probability estimates on the angles, which largely determine the convexity. – user0o Jan 17 at 20:44
@Sam: do you know about the probability that the extremal configurations in generalized happy ending problem occur for $8$ points or less? – user0o Jan 17 at 20:49
2 
 Probability distribution not clear: Are the points independent? For a given point, are the horizontal and vertical coordinates independent? – Gerald Edgar Jan 19 at 13:35
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1 Answer

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For specific and not very small $n$ this would be quit a messy computation.

There are esults about the asymptotics for large $n$ in a paper of Ambrus and Barany http://arxiv.org/abs/0906.5452 . They consider a slightly different problem, but their methods work and the result is $c n^{1/3}$ for some computable $c$.

Note also that they compute the typical value, which is only a lower bound for the expectation. Various concentration inequalities which apply since the points are iid can be used to get the expectation as well. (Or you could try to get a large deviation estimate directly.)

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Dear Omer, thank you for your answer, but do you know if the exponent would change for other convex regions? since they only considered a triangle. – user0o Jan 25 at 3:46
The exponent would not change. For example, take random points in a circular disk. Fit a triangle in the disk. A constant fraction of the n points falls inside the triangle (w.h.p.), and thus you get at least $c′n^{1/3}$ points in convex position; similarly, an ellipse (affine image of a disk) fits into any triangle, showing the other direction of the inequality. Any two convex shapes are related like this (after affine transformation). – Günter Rote Apr 28 at 23:15

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