  We show that global lower bounds to the mode volume of a dielectric resonator
can be computed via Lagrangian duality. State-of-the-art designs rely on sharp
tips, but such structures appear to be highly sub-optimal at nanometer-scale
feature sizes, and we demonstrate that computational inverse design offers
orders-of-magnitude possible improvements. Our bound can be applied for
geometries that are simultaneously resonant at multiple frequencies, for
high-efficiency nonlinear-optics applications, and we identify the unavoidable
penalties that must accompany such multiresonant structures.
