  We continue our investigation of the Gauss variational problem for infinite
dimensional vector measures associated with a condenser $(A_i)_{i\in I}$. It
has been shown in Potential Anal., DOI:10.1007/s11118-012-9279-8 that, if some
of the plates (say $A_\ell$ for $\ell\in L$) are noncompact then, in general,
there exists a vector $\mathbf a=(a_i)_{i\in I}$, prescribing the total charges
on $A_i$, $i\in I$, such that the problem admits no solution. Then, what is a
description of all the vectors $\mathbf a$ for which the Gauss variational
problem is nevertheless solvable? Such a characterization is obtained for a
positive definite kernel satisfying Fuglede's condition of perfectness; it is
given in terms of a solution to an auxiliary extremal problem intimately
related to the operator of orthogonal projection onto the cone of all positive
scalar measures supported by $\bigcup_{\ell\in L}A_\ell$. The results are
illustrated by examples pertaining to the Riesz kernels.
