One way is to use simple von Misses Fisher uniform distribution on the unit hyperspace or vector Bingham (antipodal symmetric) distribution on the sphere.

Also : https://core.ac.uk/download/pdf/82470080.pdf

Hello Noemie,

Your question is interesting, and seems to have some contradiction; or maybe I don't understand you perfectly.

Regarding the generation of points from a uniform distribution on the surface of a convex hull, 'for something simple to understand and implement' we could start with the three-dimensional case.

You can easily generate points from a uniform distribution in the unit square. Then you can project each face of the convex hull into the unit square, clip away the points outside the projected face and transform the remaining points back in the original position of each face.

Regarding 'finding sets of points lying on convex hull boundaries' there are many convex-hull algorithms in any dimension.

Kind regards,

Frank

Dear Noemie,

I don't think that the question can really be answered without posing it in a more precise manner. First and foremost would be your expectations about the meaning of a uniform distribution of points on a complex hull...

However, if you are interested in convex hulls that have been generated from some process (and aren't easily relatable functions) then my thought would be to adapt one of the convex hull finding algorithms and take a series of random walks along the boundaries from some set of known initial points. These could be made to approximate whichever expected distances and variances you decide you are looking for.

An alternate approach would be to "walk" the boundary according to a n space grid again using one of the boundary finding algorithms and randomly select grid points. These could then be perturbed to correspond with your planned distance metric and distribution.

Good luck