Inner Riesz balayage in minimum energy problems with external fields

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$ on $\mathbb R^n$, where $n\geqslant2$, $\alpha\in(0,2]$, and $\alpha<n$, we consider the problem of minimizing the Gauss functional \[\int\kappa_\alpha(x,y)\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu,\quad\text{where $f:=-\int\kappa_\alpha(\cdot,y)\,d\omega(y)$},\] $\omega$ 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 $\omega^A(\mathbb R^n)\geqslant1$. 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.