4
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.