# 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. Here is my progress.

Since every separable Banach space isometrically embeds in $\ell_\infty$ and $L_\infty$, then the case $p=\infty$ is closed.

For $q=1,\infty$ the unit ball of $\ell_q^m$ contains segments while unit balls of $\ell_p$ and $L_p$ for $p>1$ doesn't. So for $q=1,\infty$ and $p>1$ isometric embedding doesn't exist.

Since $\ell_1$ isometrically embedded in $L_1$, then $\ell_q^m$ isometrically embeded in $\ell_1$ and $L_1$. So for $q=p=1$, we have the desired embedding.

Since $\ell_q$ isometrically embeds into $L_p$ iff $1\leq p\leq q\leq 2$ or $q=p>2$, then for the same values of $p$ and $q$ we also have isometric embedding.

Thank you for taking time.

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See  MR0417756 (54 5804) 46B05 Dor, L. E. Potentials and isometric embeddings in $L_1$. Israel J. Math. 24 (1976), no. 3-4, 260–268  for a complete answer to your question.