# Distance function to the boundary and Harnack inequality

Suppose $$\Omega \subset \mathbb{R}^d$$ be a domain, and let $$\rho(x) = \mathrm{dist} (x, \partial \Omega)$$ be the distance function to the boundary of $$\Omega$$. I want to know for which domains $$\rho$$ satisfies a Harnack type inequality. Harnack inequality says that $$\sup _{x \in B} \rho (x) \leq C \inf _{x \in B} \rho(x)$$ on a ball $$B= B(a,r), a \in \Omega$$, and $$C$$ is a constant depend on $$B$$. It is known that harmonic functions satisfy Harnack inequality. Is it enough if $$\Omega$$ satisfy regularity property (e.g.,if it is a Lipschitz or a NTA-domain)? What about boundary Harnack inequality?