Geodesic
Approximation properties of finite-dimensional subspaces in~$L_1$
Sbornik. Mathematics, Tome 18 (1972) no. 1, pp. 1-14

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It is known that if a measure $\mu$ has no atoms, then the space $L_1(T,\mu)$ contains no finite-dimensional Chebyshev subspace. In the present work it is shown that an arbitrary finite-dimensional subspace $E$ in $L_1(T,\mu)$ (for which the measure has no atoms) is almost Chebyshev, i.e. the set of elements possessing nonunique best approximations in the given finite-dimensional space $E$ is of the first category. At the same time this set is everywhere dense. There is further given a characterization of elements with nonunique best approximations. Bibliography: 16 titles. 
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     author = {S. Ya. Havinson and Z. S. Romanova},
     title = {Approximation properties of finite-dimensional subspaces in~$L_1$},
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S. Ya. Havinson; Z. S. Romanova. Approximation properties of finite-dimensional subspaces in~$L_1$. Sbornik. Mathematics, Tome 18 (1972) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/SM_1972_18_1_a0/