On Non-visibility of Kobayashi Geodesics and the Geometry of the Boundary

We show that on convex domains with \(\mathcal {C}^{1,\alpha }\) ${\mathcal{C}}^{1,\alpha }$ -smooth boundary the limit set of non-visible Kobayashi geodesics are contained in a complex face. In \(\mathbb {C}^2\) ${\mathbb{C}}^2$ this implies the existence of a complex tangential line segment of non-Goldilocks point in the boundary. Conversely, we construct sequences of non-visible almost-geodesics on ( \(\mathbb {C}\) $\mathbb{C}$ -)convex domains whose boundary contains a complex tangential line segment of non-Goldilocks points in a specific direction.