# Banach embedding of finite dimensional spaces

Recall that: let $$0, then $$\ell_r$$ uniformly contains a subspace isomorphic to $$\ell_s^m$$, $$m\ge 1$$ (see [JS]).

I am wondering whether are any result for the case when $$r>s>2$$?

[Johnson, William B.; Schechtman, Gideon Embedding $$l_p^m$$ into $$l_1^m$$, Acta Math. 149 (1982), 71--85.][JS]

For $$2, if $$\ell_s^n$$ embeds uniformly into $$\ell_r$$ for all $$n$$, then either $$s=r$$ or $$s=2$$. This is basically the localization to finite dimensions of the classical dichotomy theorem of Kadec and Pelczynski. The book of Albiac and Kalton is a good source for this.