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

## TL;DR

This paper investigates the geometry of random polytopes generated by heavy-tailed vectors, showing they contain a canonical body with high probability, and applies these findings to sparse recovery in compressive sensing.

## Contribution

It introduces minimal assumptions on the generating vectors and establishes the presence of a canonical body in the polytopes, extending previous results to heavy-tailed distributions.

## Key findings

- Random polytopes contain a deterministic polar of a floating body with high probability.
- Established estimates for heavy-tailed random vectors like $q$-stable vectors.
- Applied geometric results to noise blind sparse recovery in compressive sensing.

## Abstract

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\mathbb{R}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ 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.