On Dvoretzky's theorem for subspaces of $L_p$

Grigoris Paouris, Petros Valettas

Published 2015-10-25Version 1

We prove that for any $p > 2$ and every $n$-dimensional subspace $X$ of $L_p$, the Euclidean space $\ell_2^k$ can be $(1 + \varepsilon)$-embedded into $X$ with $k \geq c_p \min\{\varepsilon^2 n, (\varepsilon n)^{2/p} \}$, where $c_p > 0$ is a constant depending only on $p$.