Bounds for capacities in terms of asymmetry

The inequality (1.3) was conjectured by L. E. Fraenkel and, as noted in [6], the exponent 2 in (1.3) is sharp. The proof in [7] relies on an inequality between capacity and moment of inertia which had been proved by Pólya and Szegö [10 ; p 126] for connected sets. For general sets, this inequality had remained open until Hansen and Nadirashvili’s ingenious proof in [7]. They also showed that, in (1.3), K1 ≥ 1/4. The proofs in [6] are based on estimates for condensers.