Measuring the Randomness and Statistics of Convex Polygons 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: 


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*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 


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*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.

 A: I write my comment as an answer because it is a bit longer.
As Joseph O'Rourke indicated, the question needs to be made more precise because it is not clear (to me) what it asks. One way of generating a random polygon is to choose $n$ points  independently according to some probability measure on $\newcommand{\bR}{\mathbb{R}}$ $\bR^2$.
For example, if you choose the  $n$ random points independently and uniformly in  a disk of radius $R$, then  an old result of Renyi and Sulanke states that, as $n\to\infty$, the expected number of vertices of their convex hull  behaves like $Z (nR^2)^{1/3}$, where $Z$ is an explicit universal constant.  In fact, for large $n$, the number of vertices of the convex hull is highly concentrated around its mean.  (This is a very  general phenomenon; see this paper  and the references therein.)
If you choose your $n$ points randomly and uniformly in the interior of a regular convex $r$-gon, then Renyi-Sulanke proved that for large $n$, the expected number of vertices of their convex hull is dramatically smaller, namely it behaves like $\frac{2r}{3}\log n$ as $n\to\infty$.   Again,the number of  vertices of this random hull is  highly concentrated around its mean.
Moreover,  in both cases, as $n\to \infty$, the  random convex hull becomes highly concentrated near a certain limiting convex region that depends on the concept of randomness you use. (In particular, no chaotic white-noise-like behavior.)
I have mentioned these results to point out that,  whatever answers you expect, they will depend on the concept of randomness you use. Hence the need to  describe this randomness precisely.
