A geometric description of domains whose Hardy constant is equal to 1/4

We give a geometric description of families of non-convex planar and spatial domains in which the following Hardy inequality holds: the Dirichlet integral of any smooth compactly supported function $f$ on the domain is greater than or equal to one quarter of the integral of $f^2(x)/\delta^2(x)$, where $\delta(x)$ is the distance from $x$ to the boundary of the domain. Our geometric description is based analytically on new one-dimensional Hardy-type inequalities with special weights and on new constants related to these inequalities and hypergeometric functions.