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We continue our investigation of the Gauss variational problem for infinite dimensional vector measures on a locally compact space, associated with a condenser ( ). It has been shown by Zorii (Potential Anal 38:397–432, ) that, if some of the plates (say   for ℓ ∈ ) are noncompact then, in general, there exists a vector  = ( ), prescribing the total charges on  ,  ∈ , such that the problem admits  solution. Then, what is a description of the set of all vectors  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 convex cone of all nonnegative scalar measures supported by ∪   . The results are illustrated by several examples pertaining to the Riesz kernels.

Necessary and Sufficient Conditions for the Solvability of the Gauss Variational Problem for Infinite Dimensional Vector Measures