Universal relation between energy gap and dielectric constant

We establish a universal relation between the energy gap and the static dielectric constant for all insulating states. This relation yields an upper bound on the energy gap, which only depends on the electron density and electronic dielectric constant. We identify two types of energy gaps associated with transverse and longitudinal excitations at long wavelength, which correspond to the optical gap and the plasmon energy respectively. Their upper bounds are set by the dielectric constant and its inverse respectively. The transverse gap bound is calculated for a wide range of materials and compared with the measured optical gap. A remarkable case is cubic boron nitride, in which the direct gap reaches \SI{72}{\percent} of the bound. Our results are derived from the Kramers-Kronig relation and the $f$-sum rule, and therefore rest on general physical principles.