The observation is made that the elastic compliance and the thermal/electric conductivity of a Hertzian contact depend on the contact geometry in a similar way. Being extended to a system of multiple contacts (a granular material, or a contact of two rough surfaces) it yields an explicit connection between the effective conductivity and the effective (incremental) elastic compliance. Being independent of the contact geometries, this cross-property connection allows one to estimate the elastic compliance from the electric conductivity data. Another observation is that this cross-property connection is very similar to the one for a material with cracks. This implies that such connections are more or less independent of the nature of discontinuity and can be applied, for example, to cracks with partial contacts between the faces or to a granular material with microcracked grains.