Near-Optimal Embeddings of Constant-Dimensional Subspaces of $L^p$ into $\ell_p^N$

Abstract

For $d\geq 2$, $p\geq 1$ and $ε> 0$, let $N_{p}(d,ε)$ be the smallest integer $N$ such that every $d$-dimensional subspace of $L^{p}[0,1]$ admits a linear embedding into $\ell_{p}^{N}$ with distortion at most $1+ε$. For fixed $d$ and $p$, the bound \[ N_{p}(d,ε) = \widetilde{O}_{d,p}\left(ε^{-2(d-1)/(d+2p)}\right) \] is established. For $p\notin 2\mathbb{Z}$, this is optimal up to logarithmic factors; for positive even integers $p$, isometric embeddings of dimension independent of $ε$ are known. The stated upper bound was previously known only for integer $p$.