Sharp constant in the Hardy inequality involving the circumradius (Menger–Melnikov curvature)

Determine the optimal (smallest) constant CH > 0 in the Hardy-type inequality ∫ℝ^2 |A[|u|^2]|^2 |u|^2 ≤ CH (∫ℝ^2 |u|^2)(∫ℝ^2 |∇|u||^2) for all u ∈ H^1(ℝ^2), equivalently the best constant controlling the Menger–Melnikov curvature ∫∫∫ ρ(x)ρ(y)ρ(z)/R(x,y,z)^2 dx dy dz (with ρ=|u|^2) by the product of mass and Dirichlet energy.

Background

A key ingredient in the paper’s upper bounds for γ*(β) is a Hardy-type inequality that bounds the magnetic self-interaction term ∫ |A[|u|^2]|^2 |u|^2 by a constant times the product of mass and the Dirichlet energy of |u|. Through an identity, this term is equal to (1/6) times the triple integral of the inverse squared circumradius (the Menger–Melnikov curvature) against ρ(x)ρ(y)ρ(z), with ρ = |u|^2.

While a bound with constant 3/2 is available, the sharp constant is not known. The authors explicitly note that the optimal constant for the Hardy inequality involving the circumradius has not been determined, and improving it would sharpen upper bounds on γ*(β).