On the geometry of random convex sets between polytopes and zonotopes

In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a qth-moment (q≥1) of an average of order statistics of 1-dimensional marginals of a sequence of N≥n independent random vectors in Rn. We consider the random model of isotropic log-concave distributions as well as the uniform distribution on an ℓnp-sphere (1≤