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In this paper we consider eigenfunctions of the Laplacian on a planar domainwith polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions.The main result is a quantitative estimate on the $L^2$ mass of eigenfunctionsnear a point in terms of the distance to the nearest non-adjacent boundaryface. In particular, eigenfunctions cannot concentrate completely at any onesingle point. The technique of proof is to use the commutator ideas from therecent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.

A one point non-concentration estimate for Laplace eigenfunctions on polygons