Eigenfunction concentration for polygonal billiards

In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$\int_U |u|^2 \geq c \int_B |u|^2$$ for some $c = c(U) > 0$ and any eigenfunction $u$.