The pluricomplex Poisson kernel for convex finite type domains

Given a bounded convex domain $D\subset {\mathbb{C}}^n$ of finite D’Angelo type and a boundary point $\xi \in \mathrm{\partial }D$, we prove that the homogeneous complex Monge-Ampère equation $(d{d}^{c}u{)}^n=0$ possesses a continuous strictly negative solution ${\mathrm{\Omega }}_{\xi }$ that vanishes on $\mathrm{\partial }D\setminus \{\xi \}$ and has a simple pole at $\xi$. We establish that ${\mathrm{\Omega }}_{\xi }(z)$ equals (up to sign) the normal derivative at $\xi$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $\xi$. Moreover, ${\mathrm{\Omega }}_{\xi }$ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, ${\mathrm{\Omega }}_{\xi }$ serves as a generalisation of the classical Poisson kernel of the unit disc. Our approach, based on metric methods and scaling techniques, allows our results to be applied to strongly convex domains with $C^2$-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.