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