Equivalence and self-improvement of p-fatness and Hardy's inequality, and association with uniform perfectness

We present an easy proof that p-Hardy's inequality implies uniform p-fatness of the boundary when p = n. The proof works also in metric space setting and demonstrates the self-improving phenomenon of the p-fatness. We also explore the relationship between p-fatness, p-Hardy inequality, and the uniform perfectness for all p ≥ 1, and demonstrate that in the Ahlfors Q-regular metric measure space setting with p = Q, these three properties are equivalent. When p ≠ 2, our results are new even in the Euclidean setting.