Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures

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We continue our investigation of the Gauss variational problem for infinite dimensional vector measures associated with a condenser (Aᵢ)ᵢ∈ ᵢ. It has been shown in Potential Anal., DOI:10.1007/s11118-012-9279-8 that, if some of the plates (say Aℓ for ℓ∈ L) are noncompact then, in general, there exists a vector a=(aᵢ)ᵢ∈ ᵢ, prescribing the total charges on Aᵢ, i∈ I, such that the problem admits no solution. Then, what is a description of all the vectors 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 ℓ∈ ₗAℓ. The results are illustrated by examples pertaining to the Riesz kernels.