  For every $n\geq 3,$ we construct an $n$-dimensional Banach space which is
isometric to a subspace of $L_{1/2}$ but is not isometric to a subspace of
$L_1.$ The isomorphic version of this problem (posed by S. Kwapien in 1969) is
still open. Another example gives a Banach subspace of $L_{1/4}$ which does not
embed isometrically in $L_{1/2}.$ Note that, from the isomorphic point of view,
all the spaces $L_q$ with $q<1$ have the same Banach subspaces.
