How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)?

What is known about the generation of random convex polygons (or polyhedra)?

In view of the comments, I will address the choice of method of generating convex polygons:

Take the convex hull of a set of points, that are randomly selected from a convex region; that method appears to be the one, which is understood the best.

Starting with a simplex whose corners are randomly selected points, repeatedly cut off a single vertex by a hyperplane that separates that vertex from the rest, until the desired number of vertices is reached.

repeatedly and randomly select $n$ points from a random set, until a set in convex configuration is encountered.

While it seems that only the first method is well understood, especially the second method seems suitable for efficiently generating large sets of convex polygons with a predefined number of vertices.

As for the statistical properties, I could imagine

distribution of angles between adjacent edges

distribution of edge lengths

distribution of polygon areas

distribution of diameter modality

There are a lot of tests for checking the randomness of a sequence of numbers, e.g. George Marsaglia's Diehard test battery (cf e.g. http://en.wikipedia.org/wiki/Diehard_tests) and I wonder if something in that line could also be devised for a sequence convex polygons.