Conditional Chebyshev center of a bounded set of continuous functions

Subspaces $\{\mathscr L^n\}$ of codimension $n<\infty$ of the space $C(T)$ of functions, continuous in a bicompactum $T$, are considered. A criterion, whereby a subspace $\mathscr L^n$, contains a Chebyshev center for any bounded set of $C(T)$, is established in terms of the properties of the supports of measures which are annihilated in $\mathscr L^n$. This criterion is equivalent to the following conditions: $\mathscr L^n$ contains an element of best approximation for every $x\in C(T)$, and the support of every measure, which is annihilated in $\mathscr L^n$, is extremally unconnected with respect to the bicompactum $T$.