In this paper we study geodetic convex hulls in graphs and prove new bounds on the size of hull sets and related parameters. A set of vertices in a graph is geodetic convex if no shortest path between vertices in the set leaves the set. The geodetic hull of a vertex set is its smallest convex superset. A hull set is a set of vertices whose geodetic hull is the entire graph. Computing a hull set of minimum size poses an NP-hard problem. We present an exact kernelization algorithm which implies that the problem is FPT in the vertex cover number. Furthermore we provide new bounds on the size of a minimum hull set in relation to other graph parameters. For practical application, we design an improved heuristic as well as an algorithm to obtain strong instance-based lower bounds. We evaluate the efficiency and quality of the heuristic and the lower bound on diverse graphs. Our experiments demonstrate that our methods outperform previous ones by a large margin.