The problem of calculating the effective dielectric constant of a composite material εe is analyzed by separating out the dependence of εe on the microscopic geometry. A set of characteristic geometric functions is defined, which depend in detail on the microscopic geometry of the material under discussion, but whose general analytical properties can be derived. The pole structure of these functions is shown to have important consequences. It is experimentally observable in certain optical and microwave experiments on metal-insulator composites. Furthermore, it can be used to obtain a considerable amount of quantitative information about εe in the form of rigorous upper and lower bounds. Finally, it allows a new approach to be taken in the discussion of the critical properties of composite systems near a percolation threshold. The characteristic functions defined and discussed in this article are applicable not only to εe, but also to other scalar material constants of the composite material, such as the magnetic permeability, electrical and thermal conductivity, and diffusivity.