Proximinal Subspaces of C(Q) of Finite Codimension

In this paper we investigate the structure of a proximinal subspace G of C(Q) of codimension n, in terms of the geometry of the range of the vector measure ν=(ν₁,...,ν_n), where (ν₁,...,ν_n) is a basis for the annihilator G^⊥. In particular, we prove that if ν is non-atomic, G is proximinal iff for every P∈ExtR(ν) there exists a clopen subset C of ∪ⁿ_i=1S(ν_i) such that ν(C)=P.