For the Riesz kernel \(\kappa _\alpha (x,y):=|x-y|^{\alpha -n}\) on \({\mathbb {R}}^n\) , where \(n\geqslant 2\) , \(\alpha \in (0,2]\) , and \(\alpha <n\) , we consider the problem of minimizing the Gauss functional \(\begin{aligned} \int \kappa _\alpha (x,y)\,d(\mu \otimes \mu )(x,y)+2\int f\,d\mu ,\quad \,{\text {where }}\,f:=-\int \kappa _\alpha (\cdot ,y)\,d\omega (y), \end{aligned}\) \(\omega \) being a given positive (Radon) measure on \({\mathbb {R}}^n\) , and \(\mu \) ranging over all positive measures of finite energy, concentrated on \(A\subset {\mathbb {R}}^n\) and having unit total mass. We prove that if A is a quasiclosed set of nonzero inner capacity \(c_*(A)\) , and if the inner balayage \(\omega ^A\) of \(\omega \) onto A is of finite energy, then the solution \(\lambda _{A,f}\) to the problem in question exists if and only if either \(c_*(A)<\infty \) , or \(\omega ^A({\mathbb {R}}^n)\geqslant 1\) . Despite its simple form, this result improves substantially some of the latest ones, e.g. those by Dragnev et al. (Constr. Approx., 2023) as well as those by the author (J. Math. Anal. Appl., 2023). We also provide alternative characterizations of \(\lambda _{A,f}\) , and analyze its support. As an application, we show that if A is not inner \(\alpha \) -thin at infinity, and is "very thin" at \(z\in \partial _{{\mathbb {R}}^n}A\) (to be precise, if \(c_*(A^*_z)<\infty \) , \(A^*_z\) being the inverse of A with respect to the unit sphere centered at z), then the above problem with \(\omega :=q\varepsilon _z\) , where \(q\in [1,\infty )\) and \(\varepsilon _z\) denotes the unit Dirac measure at z, is still solvable. Thus no compensation effect occurs between the two oppositely signed charges, \(-q\varepsilon _z\) and \(\lambda _{A,f}\) , carried by the same conductor A, which seems to contradict our physical intuition.
