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.



Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
See 
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MR0417756 (54 5804) 46B05 Dor, L. E. Potentials and isometric embeddings in $L\_1$. Israel J. Math. 24 (1976), no. 3-4, 260–268
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for a complete answer to your question.


