Convexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.