We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is a norm with a polytope unit ball. The condition is given by a geometric criterion involving how the row span of the design matrix intersects the faces of the dual norm unit ball. The criterion also provides information about the model selection properties of the corresponding estimation method. For this, our analyses cover LASSO, the related method of basis pursuit, as well as the SLOPE estimator.