We characterize the essential spectrum of the plasmonic problem for polyhedra in $mathbb{R}^3$. The description is particularly simple for convex polyhedra and permittivities $epsilon < - 1$. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann--Poincare operator. We therefore study the spectral structure of the the double layer potential for polyhedral cones and polyhedra.
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