A Note about Torsional Rigidity and Euclidean Moment of Inertia of Plane Domains

Abstract

Denote by P(G) the torsional rigidity of a simply connected plane domain G, and by I₂(G) the Euclidean moment of inertia of G. In 1995 F.G. Avkhadiev proved that P(G) and I2(G) are comparable quantities in sense of Pólya and Szegö. Moreover, it was shown that the ratio P(G) /I₂(G) belongs to the segment [1, 64]. We investigate the following conjecture P(G) ≥ 3I₂(G), where G is a simply connected domain. We prove that the conjecture is true for polygonal domains circumscribed about a circle. For convex domains we show sharp isoperimetric inequalities, which justify the conjecture, in particular, we prove that P(G) > 2I₂(G). Some aspects of approximate formulas for P(G) are also discussed.