We consider, and study with elementary calculus, the polyhedral norms ||x||(k)= sum of the k largest among the |xi|'s. Besides their basic properties, we provide various expressions of the unit balls associated with them and determine all the facets and vertices of these balls. We do the same with the dual norm ||.||∗(k) of ||.||(k). The study of these polyhedral norms is motivated, among other reasons, by the necessity of handling sparsity in some modern optimization problems, as is explained at the end of the paper.