[R] sampling from the unoform distrubuton over a convex hull

[Alberto Vieira Ferreira Monteiro]

Duncan Murdoch wrote:
>
> Suggestion:  Find a rectangular region containing the hull, and sample
> uniformly there.  Accept points that don't expand the hull of the
> original points.
>
This is a feasible solution only in lower dimensions; the area of the
convex hull can become exponentially small relative to its external
rectangle. For example, if the convex hull is approximately spherical,
the relation of the (hyper)volumes are (time to summon Wikipedia)...
http://en.wikipedia.org/wiki/Hypersphere#Hyperspherical_volume
...

Volume.ratio <- pi^(n/2) * 2^(-n) / gamma(n/2 + 1)

Even for n = 5 this would be 0.16

The hypervolume of a _hyperpyramidal_ convex hull would be even
worse, as it would be 1/n! (= 1/gamma(n+1)) of the hypervolume of the
hypercube.

Alberto Monteiro