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

Abstract

We study the geometry of centrally symmetric random polytopes, generated by [Formula: see text] independent copies of a random vector [Formula: see text] taking values in [Formula: see text]. We show that under minimal assumptions on [Formula: see text], for [Formula: see text] and with high probability, the polytope contains a deterministic 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 were obtained previously, and thanks to the minimal assumptions on [Formula: see text], we derive estimates in cases that were out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when [Formula: see text] is [Formula: see text]-stable or when [Formula: see text] has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing — noise blind sparse recovery.