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A Banach subspace of $L_{1/2}$ which does not embed in $L_1$ (isometric version)

Alexander Koldobsky
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Abstract

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.

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Alexander Koldobsky. 1994-04-20. A Banach subspace of $L_{1/2}$ which does not embed in $L_1$ (isometric version). https://arxiv.org/abs/math/9404212

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