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

We continue our investigation of the Gauss variational problem for infinite dimensional vector measures on a locally compact space, associated with a condenser (A (i) ) (i aaEuro parts per thousand I) . It has been shown by Zorii (Potential Anal 38:397-432, 2013) that, if some of the plates (say A (a"") for a""aEuro parts per thousand aaEuro parts per thousand L) are noncompact then, in general, there exists a vector a = (a (i) ) (i aaEuro parts per thousand I) , prescribing the total charges on A (i) , i aaEuro parts per thousand I, such that the problem admits no solution. Then, what is a description of the set of all 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 convex cone of all nonnegative scalar measures supported by a(a)aEuro parts per thousand(a""aEuro parts per thousand aaEuro parts per thousand L) A (a""). The results are illustrated by several examples pertaining to the Riesz kernels.