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We revisit the phenomenon where, for certain domains $D$, if the squeezingfunction $s_D$ extends continuously to a point $p\in \partial{D}$ with value$1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$,we present weaker conditions under which the latter conclusion is obtained. Inanother direction, we show that there are bounded domains $D\Subset\mathbb{C}^n$, $n\geq 2$, that admit large $\partial{D}$-open subsets$\mathscr{O}\subset \partial{D}$ such that $s_D\to 0$ approaching any point in$\mathscr{O}$. This is impossible for planar domains. We pose a few questionsrelated to these phenomena. But the core result of this paper identifies a newfamily of holomorphic homogeneous regular domains. We show via a family ofexamples how abundant domains satisfying the conditions of this result are.

A new family of holomorphic homogeneous regular domains and some questions on the squeezing function