Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in ${\mathbb{C}\mathbb{P}}^n$

In this paper, we use Takeuchi’s Theorem to show that for every Lipschitz pseudoconvex domain $\mathrm{\Omega }$ in ${\mathbb{C}\mathbb{P}}^n$ there exists a Lipschitz defining function $\rho$ and an exponent $0<\eta <1$ such that $-(-\rho {)}^{\eta }$ is strictly plurisubharmonic on $\mathrm{\Omega }$. This generalizes a result of Ohsawa and Sibony for $C^2$ domains. In contrast to the Ohsawa-Sibony result, we provide a counterexample demonstrating that we may not assume $\rho =-\delta$, where $\delta$ is the geodesic distance function for the boundary of $\mathrm{\Omega }$.