I know that this question is not well defined, but some clusters tend to be elliptical or lie in lower dimensional space whilst the other have nonlinear shapes (in 2D or 3D examples).
Is there any measure of nonlinearity (or "shape") of clusters?
Note that in 2D and 3D space, it is not a problem to see the shape of any cluster, but in higher dimensional spaces it is problem to say something about shape. In particular, are there any measures of how convex cluster is?
I was inspired for this question by many other clustering questions where people talk about clusters but nobody is able to see them (in higher dimensional spaces). Moreover, I know that there are some measures of nonlinearity for 2D curves.