In the literature we know that the eigenvalue of Dirichlet Laplacian are defined on an open set of $\mathbb{R}^n$ but there are times I fall on paper which define it on closed or even compact sets. For example: https://www.sciencedirect.com/science/article/pii/S0001870804001732 page 107 equality (3), he shows an inequality or equality for the first eigenvalues of Dirichlet Laplacian among convex bodies.
Did they make a mistake or is it me who makes the mistake.
The Laplacian eigenvalues are defined for open sets. In the paper the authors consider "the family of $n$-dimensional convex bodies, i.e. compact, convex subsets of $\mathbb{R}^n$, with non-empty interior". When they talk about eigenvalues of such a set $K$, they mean the eigenvalues of the interior of $K$.