The Helly problem and best approximation of summable functions

Approximative properties are studied for the subspaces $\{L^n\}$ of finite codimensionality $n$ in the space of summable functions $L_1=L_1(T,\Sigma,\mu)$. Criteria are established for a subspace in which for every $x\in L_1$ there exists an element of best approximation (or a unique such element). A dual interpretation of these results in terms of the existence and uniqueness of minimal solutions of the finite problem of moments (“Helly problem”) is given. The question of construction of generalized elements of best approximation in a certain extension of the space $L_1$ is considered.
Bibliography: 18 titles.