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 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
