Improved bounds for Laue's constant and multivariate extensions

Denote by P the class of probability density functions on ℝ with a nonnegative and integrable characteristic function. For each p ∈ P, there exists an adjoint density p̂, which is proportional to the characteristic function of p. The products λ(p) = Var(p)Var(p̂) have a greatest lower bound Λ, known as Laue's constant. In this paper we improve the previous estimates of Λ, proving that 0.543 < Λ < 0.85024. Further results and related estimates are also obtained for a multivariate version of the uncertainty relation which is based on univariate projections.