We will start by explaining how to embed a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n$, to globally expose a given point on the boundary. The motivation comes from a question about existence of a positive squeezing radius. The technique is an adaption of techniques for exposing points on boundaries of Riemann surfaces in $\mathbb{C}^2$ due to Forstneric-Wold. We will then proceed to explain how the techniques can be used to construct smooth parameter families of global support surfaces for convex domains of finite 1-type.
This is joint work with Fornaess and Diederich.