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}.
Categories: math.AP math.FA math.SP