I need to deal with the following. Let $[B]$ denote the set of integers $\{0,1,\ldots,B\}$.

Consider an integer $n \geq 3$ and a grid $[B]\times [B]$, and let $P_{n,B}$ denote the set of convex polytopes in $[B]\times [B]$ of $n$ vertices. How do we sample a uniform element in $P_{n,B}$?

In other words, I need an algorithmic to uniquely generate all the convex polytopes of a given number of vertices.

For instance, if $n=3$, then we can simply sample any three different points in $[B]\times[B]$, since they automatically define a convex polytope of three vertices. However, for $n>3$ this does not work since $n$ points may not define a convex polytope.

I assumed this problem has been studied before. Anyone has some insights?

Thanks!

  • I assume you've considered rejection sampling? Pick any $n$ distinct points, take their convex hull, accept it if it has $n$ vertices, otherwise try again. Note that you'll have to do this even when $n=3$, since the points may be collinear. – Rahul Oct 11 at 12:11
  • @Rahul Thanks Rahul. Definitely, that's our standard approach, but it scales very badly since, when you increase $n$, the probability of getting a convex hull with a large number of nodes (relatively close to $n$) goes down rather quickly. (Thanks for pointing out the issue with $n=3$ :-) ) – Daniel Oct 11 at 12:12
  • Have you searched some of these links? google.com/… ? The requirement for grid points might be the stumbling block. – Ethan Bolker Oct 11 at 14:23

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