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arXiv · 1510.07289

On Dvoretzky's theorem for subspaces of $L_p$

Grigoris Paouris·Petros Valettas
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Abstract

We prove that for any $2 \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ α_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0<\varepsilon<1 , \] where $Z$ is a standard $n$-dimensional Gaussian vectors, $α_p>0$ is a constant depending only on $p$ and $C,c>0$ are absolute constants. As a consequence we show optimal lower bound for the dimension of almost spherical sections for these spaces. In particular, for any $2 0$ is a constant depending only on $p$. This improves upon the previously known estimate due to Figiel, Lindenstrauss and V. Milman.

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Grigoris Paouris, Petros Valettas. 2017-10-20. On Dvoretzky's theorem for subspaces of $L_p$. https://arxiv.org/abs/1510.07289

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