Approximation properties of finite-dimensional subspaces in $L_1$

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.
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