# Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.

Since $\ell_q^m$ is finite dimensional some (not necessary isometric) embedding always exist. I'm interested in isometric ones.

Thank you for taking time.

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It's not exactly addressing your question, but did you see this blog post by Leonid Kovalev? He recommends the book by Milman and Schechtman for the "real stuff". –  commenter Oct 29 '12 at 21:00
what does "real stuff" mean? –  Norbert Oct 29 '12 at 21:02
I'm only quoting him. I suppose he means that the sharpest known embedding constants between finite-dimensional $\ell_p$-spaces and much more can be found there. I don't know the book, but given the reputation of its authors it's probably worth a look. // You said elsewhere that you were reading the book by Albiac-Kalton. The chapter on local theory should contain something, too. –  commenter Oct 29 '12 at 21:05
Thank you @commenter –  Norbert Oct 29 '12 at 21:07
@commenter Yes using result on embedding of $\ell_q$ in $L_p$ we can get some partial progress. The case $p=\infty$ is also of small interest because every separeable space ismotrically embedded in $\ell_\infty$ and $L_\infty$. –  Norbert Oct 29 '12 at 21:08