On the geometry of polytopes generated by heavy-tailed random vectors

We study the geometry of centrally-symmetric random polytopes, generated by N independent copies of a random vector X taking values in R n. We show that under minimal assumptions on X, for N n and with high probability, the polytope contains a determin-istic set that is naturally associated with the random vector-namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing-noise blind sparse recovery