  We consider a type of Hardy-Sobolev inequality, whose weight function is
singular on the whole domain boundary. We are concerned with the attainability
of the best constant of such inequality. In dimension two, we link the
inequality to a conformally invariant one using the conformal radius of the
domain. The best constant of such inequality on a smooth bounded domain is
achieved if and only if the domain is non-convex. In higher dimensions, the
best constant is achieved if the domain has negative mean curvature somewhere.
If the mean curvature vanishes but is non-umbilic somewhere, we also establish
the attainability for some special cases. In the other direction, we also show
the best constant is not achieved if the domain is sufficiently close to a ball
in $C^2$ sense.
