The complex effective dielectric constant of a composite medium &egr;eis a function of the complex dielectric constants of the homogeneous components &egr;Pdi which depends also on the microscopic geometry. A characteristic geometric function is introduced to describe this dependence for the case of &egr;eand other, similar material constants, and its general analytical properties are derived and discussed. A useful representation is found for this function and we show how it may be used to derive rigorous bounds on &egr;ein various situations. The characteristic function is also used to discuss the behavior of &egr;ein a disordered conductor‐dielectric mixture near the percolation threshold of the conductor. It is found that Re &egr;ediverges as the threshold is approached from either side. At the threshold itself, both Re&egr;eand Im&egr;eare shown to acquire a peculiar frequency dependence.