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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 $ℝn$. 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 were obtained previously, and thanks to the minimal assumptions on $X$, we derive estimates in cases that were 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.