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For every      n ≥  3 , n\geq 3,       we construct an      n n      -dimensional Banach space which is isometric to a subspace of      L 1 / 2 L_{1/2}       but is not isometric to a subspace of      L 1 . 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 L_{1/4}       which does not embed isometrically in      L 1 / 2 . L_{1/2}.       Note that, from the isomorphic point of view, all the spaces      L q L_q       with      q &gt; 1 q&gt;1       have the same Banach subspaces.

A Banach subspace of 𝐿_{1/2} which does not embed in 𝐿₁ (isometric version)