On non-visibility of Kobayashi geodesics and the geometry of the boundary

We show that on convex domains with sufficiently smooth boundary the limit set of non-visible Kobayashi geodesics are contained in a complex face. In two dimensions, this implies the existence of a complex tangential line segment of non-Goldilocks points in the boundary. Conversely, we construct sequences of non-visible almost-geodesics on convex domains whose boundary contains a complex tangential line segment of non-Goldilocks points in a specific direction.