An exact sum rule is derived that links the structure of fluids adsorbed in wedges and at edges to the interfacial free energy far from the wedge apex. By focusing on hard-wall models, one observes a correspondence between interfacial statistical mechanics and geometry. The physical necessity of this correspondence can be argued from the presence of complete drying at a hard wall. Invoking the potential distribution theorem generates yet another class of geometric results, this time concerning the excluded volume generated by a sphere rolling along the surface of the wedge. Direct proof of these latter geometric theorems is straightforward in two-dimensions. Acute wedges and the right-angled wedge, provide examples of models for which comparison with simulation data and density functional theory are available.