A note on the uniformity of the constant in the Poincar\'e inequality

The classical Poincar\'e inequality establishes that for any bounded regular domain $\Omega\subset \R^N$ there exists a constant $C=C(\Omega)>0$ such that $$ \int_{\Omega} |u|^2\, dx \leq C \int_{\Omega} |\nabla u|^2\, dx \ \ \forall u \in H^1(\Omega),\ \int_{\Omega} u(x) \, dx=0.$$ In this note we show that $C$ can be taken independently of $\Omega$ when $\Omega$ is in a certain class of domains. Our result generalizes previous results in this direction.