We report here for the first time on surface immobilization of hollow faceted polyhedrons formed from catanionic surfactant mixtures. We find that electrostatic interaction with the substrate dominates their adhesion behavior. Using polyelectrolyte coated surfaces with tailored charge densities, polyhedrons can thus be immobilized without complete spreading, which allows for further study of their mechanical properties using AFM force measurements. The elastic response of individual polyhedrons can be locally resolved, showing pronounced differences in stiffness between faces and vertexes of the structure, which makes these systems interesting as models for structurally similar colloidal scale objects such as viruses, where such effects are predicted but cannot be directly observed due to the smaller dimensions. Elastic constants of the wall material are estimated using shell and plate deformation models and are found to be a factor of 5 larger than those for neutral lipidic bilayers in the gel state. We discuss the molecular origins of this high stiffness.