  In this paper, we prove that the distance function of an open connected set
in $\mathbb R^{n+1}$ with a $C^{2}$ boundary is superharmonic in the
distribution sense if and only if the boundary is {\em weakly mean convex}. We
then prove that Hardy inequalities with a sharp constant hold on {weakly mean
convex} $C^{2}$ domains. Moreover, we show that the {weakly mean convexity}
condition cannot be weakened. We also prove various improved Hardy inequalities
on mean convex domains along the line of Brezis-Marcus \cite{BM}.
