Let $\Omega$ be a bounded domain of $\mathbb{R}^N$ with smooth boundary. Show that ${\rm dist}(x,\partial \Omega)^{-\vartheta} \in L^1(\Omega)$ if $\vartheta \in (0,1)$.
I can see why if $\Omega$ is a ball centred at the origin. I don't know how to show it though if it isn't (or indeed, if the assumption that $\Omega$ has a smooth boundary is necessary).
Edit: just some minor clarifications
Cover the boundary of the domain with finitely many balls such that within each, the piece of the boundary is the graph of a smooth function with bounded derivative in some coordinate system. (This is possible by the definition of a smooth boundary).
Within such a ball, the boundary is $w_N = \phi(w_1,\dots,w_{N-1})$. Apply the straightening transformation $\zeta = (w_1,\dots,w_{N-1})-\phi(w_1,\dots,w_{N-1})$. Observe that this is a bi-Lipschitz map: the inverse of $w=w(\zeta)$ is a similar map with $+$ instead of $-$, so both of them satisfy the Lipschitz condition.
A bi-Lipschitz change of variables contributes a bounded factor to the Jacobian in the integral, and it changes the distances between points and sets at most by some factor. Thus, the problem reduces to showing that the integral of $\zeta_N^{-\theta}$ within some bounded set is finite; this is trivial.