The Levi problem for $j$-metric balls of a proper pseudoconvex domain in $\mathbb C^n$

Let $\Omega _j(a,r)$ be an $a$-centered $j$-metric ball of a proper pseudoconvex domain $\Omega $ in $\mathbb C^n$, with radius $r \gt 0$. In this paper, we discuss whether $\Omega _j(a,r)$ can be pseudoconvex and so can be holomorphically convex and vice versa. We study three principal cases of the domain $\Omega $ and we provide in each case optimal conditions on $a$ and $r$ for which the original Levi problem can be solved in $\Omega _j(a,r)$. As an application, we show that Kiselman’s minimum principle can hold in the $j$-metric setting.