Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex domain and let $p \in \partial D$ be a strongly pseudoconvex boundary point . Often it is useful to localize the boundary of $D$ near the strongly pseudoconvex point $p$ by a strongly pseudoconvex domain. That is one, would like to find a domain $G \subset D$ that is smoothly bounded and strongly pseudoconvex such that for some $\delta > 0$, $$ \partial D \cap \mathbb{B}(p, \delta) = \partial G \cap \mathbb{B}(p, \delta). $$
I believe the way to construct $G$ is as follows. Since $D$ is strongly pseudoconvex, it admits a strictly plurisubharmonic defining function $r$. Let $\chi_{\epsilon}:\mathbb{R} \to \mathbb{R}$ be a smooth function which is identically zero on $(-\infty, \epsilon]$ and strongly convex and increasing on $(\epsilon, \infty)$. Define $$ \rho(z) = r(z) + \chi_{\epsilon}(|z - p|). $$ and $G$ be defined to be the set where $\rho < 0$. As $\rho$ is the sum of a strictly plurisubharmonic function and a strongly convex function, it is strictly plurisubharmonic near $p$ and the boundary of $G$ near $p$ agrees with the boundary of $D$ near $p$. It remains to show that for a suitable $\epsilon > 0$ that the $\nabla \rho \neq 0$ on $\partial G$. Does anyone know how to do this last part?