  We establish rigorous inequalities between different electronic properties
linked to optical sum rules, and organize them into weak and strong bounds on
three characteristic properties of insulators: electron localization length
$\ell$ (the quantum fluctuations in polarization), electric susceptibility
$\chi$, and optical gap $E_{\rm G}$. All-electron and valence-only versions of
the bounds are given, and the latter are found to be more informative. The
bounds on $\ell$ are particularly interesting, as they provide reasonably tight
estimates for an ellusive ground-state property -- the average localization
length of valence electrons -- from tabulated experimental data: electron
density, high-frequency dielectric constant, and optical gap. The localization
lengths estimated in this way for several materials follow simple chemical
trends, especially for the alkali halides. We also illustrate our findings via
analytically solvable harmonic oscillator models, which reveal an intriguing
connection to the physics of long-ranged van der Waals forces.
