  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 $u$.
