Hardy inequalities on mean convex domains

I will report a recent joint work with Roger Lewis and Yanyan Li. In this work, we prove that Hardy inequalities with a sharp constant hold on weakly mean convex 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. I will also outline several related open questions.