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arXiv · 1807.02396

The isotropic constant of random polytopes with vertices on convex surfaces

Joscha Prochno·Christoph Thäle·Nicola Turchi
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Abstract

For an isotropic convex body $K\subset\mathbb{R}^n$ we consider the isotropic constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$ independent random points which are distributed according to the cone probability measure on the boundary of $K$. We show that with overwhelming probability $L_{K_N}\leq C\sqrt{\log(2N/n)}$, where $C\in(0,\infty)$ is an absolute constant. If $K$ is unconditional we argue that even $L_{K_N}\leq C$ with overwhelming probability. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respectively, and, in the unconditional case, on a new $\psi_2$-estimate for linear functionals with respect to the cone measure in the spirit of Bobkov and Nazarov, which might be of independent interest.

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Joscha Prochno, Christoph Thäle, Nicola Turchi. 2018-07-06. The isotropic constant of random polytopes with vertices on convex surfaces. https://arxiv.org/abs/1807.02396

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