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.

中文翻译：

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重尾随机向量生成的多面体几何

我们研究中心对称随机多面体的几何形状，由ñ随机向量的独立副本X取值ℝn. 我们证明了在最小假设下X， 为了ñ ≳ n并且很有可能，多面体包含一个与随机向量自然相关的确定性集合——即某个浮体的极坐标。这解决了长期存在的问题，即这种随机多面体是否包含规范体。此外，通过识别与各种随机向量相关的浮体，我们恢复了之前获得的估计，这要归功于对X，我们在无法触及的情况下得出估计值，涉及由重尾随机向量生成的随机多面体（例如，当X是q-稳定或何时X具有无条件结构）。最后，结构结果用于研究压缩感知中的一个基本问题——噪声盲稀疏恢复。