Given a finite set of $n$ points $S$ in $R^2$, its convex hull, $cvx(S)$, can be obtained with the aid of many algorithms. To numerically compare these algorithms and study their complexity I need to start with an $S$. The question is what is a suitable starting set $S$.
For example if $S$ is chosen randomly from points in the unit square then as $n$ increases the set $S$ fills the square and its convex hull is approximately the square itself. Hardly a random shape. Same thing will happen if $S$ is chosen from any other given shape. It is not clear that the asymptotics obtained from such an $S$ is representative for an arbitrary starting $S$.
In this context what is a procedure for generating a random finite set $S$?
Is the following an acceptable procedure?
Start with $P_1=(0,0)$ then for $i$ from $2$ to $n$ obtain the remaining points as
$P_i=P_{i-1}+ R(\cos (2\pi t),\sin(2 \pi t))$ where at each turn $R$ and $t$ are new random numbers in $[0,1]$. This would be a random walk with random step-length and direction.
Any finite set, subject to a shift and a scale, can be generated this way. The question that remains is that do they occur with equal likelihood.