Letdenote the class of probability density functions onwith a nonnegative and integrable characteristic function. To each, there is an adjoint density, which is proportional to the characteristic function ofp. The productshave a greatest lower bound Λ, and it is known that 0.5276 < Λ < 0.8571. Several approaches to sharpen these bounds are discussed. In particular, a variational problem is considered, in whichpis supposed to have a certain compactly supported convolution root, and which leads to an improved upper estimate, Λ < 0.8567 … The paper closes with a proposal for a multivariate analogue of the uncertainty relation.