A classification theorem for planar distributions based on the shape statistics of independent tetrads

This is the authors' abstract. We don't add key points for this paper.

The uniformity of a certain shape density of a random ΠD tetrad relative to Lebesgue measure along part of the set corresponding to alignment is shown to be sufficient under certain technical conditions for the generating distribution to be the uniform distribution on a compact convex set. A counter-example is provided to show that this result fails for random IID triangles. The main theorem is used to characterize circular, elliptical and triangular uniform generators from the shape distribution of random IID tetrads.