  For $n\geq 2, p<2$ 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
