Common subspaces of $L_p$-spaces

Abstract

For $n\ge 2,p<2$ and $q>2,$ does there exist an $n$-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both $L_p$ and $L_q$? Generalizing the construction from the paper "Zonoids whose polars are zonoids" by R.Schneider we give examples of such spaces. Moreover, for any compact subset $Q$ of $(0,\infty )\setminus \{2k,k\in N\},$ we can construct a space isometric to subspaces of $L_q$ for all $q\in Q$ simultaneously. This paper requires vanilla.sty