Let \(P\) be an \(n\)-dimensional compact, convex polyhedron in \(\mathbb{R}^n\) containing the origin in its interior. The authors define the \(n\)-dimensional polyhedral Fejér kernel either as the arithmetic mean of the Dirichlet kernel, or as (\(1/\text{card}(NP\cap \mathbb{Z}^n)\) times) the square of the Dirichlet kernel. They study the almost everywhere convergence of polyhedral Fejér type means and prove positive results in the case of the \(n\)-dimensional Euclidean space and torus. On the other hand, as they show, these results cannot be extended to compact Lie groups in general.