A space-time satisfies $\mathcal{R} \geq K $ if the sectional curvatures are bounded below by $K$ for spacelike planes and above by $K$ for timelike planes (similarly, a space-time satisfies $\mathcal{R} \leq K$ if the aforementioned inequalities are reversed). We demonstrate that these curvature bound conditions together with convex functions areeffective means to study the geometry of space-times.Chapter 3 explores the relation between convex functions and geodesic connectedness of space-times. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance, a null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. In particular, timelike strictly convex hypersurfaces of Minkowski space (which are prototypical examples of space-times satisfying $\mathcal{R} \geq 0$) are geodesically connected.Chapter 4 explores the relationship between so-called $\lambda$-convex functions ($ \hess f(x,x) \geq \lambda \langle x,x \rangle $), curvature bounds, and trapped submanifolds. We show that certain types of trapped submanifolds can be ruled out for domains of space-times satisfying $\mathcal{R} \leq K$. Using the full curvature bound condition $\mathcal{R} \leq K$ allows us to extend previous results that use timelike sectional curvature bounds to rule out trapped submanifolds in the chronological future of a point.
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