On null and causal pseudoconvex space-times

The condition of pseudoconvexity has important implications for geodesic structures of space-time. Low showed that this condition guarantees the existence of a smooth structure of the space of geodesics. Recently, the place of this property within the causality ladder is in the attention. There is a conjecture which states that every strongly causal space-time is causally simple if and only if it is null pseudoconvex. In this paper, we study the difference of causal and null pseudoconvexity in space-times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity. Then, we show that in the case of open subspaces of n-Minkowski space-time, this is causal geodesic connectedness. Finally, we prove a refined form of the conjecture that says every strongly causal space-time is causally simple if and only if it is maximal null pseudoconvex.