  We provide a necessary and sufficient condition for the uniqueness of
penalized least-squares estimators whose penalty term is given by a norm with a
polytope unit ball, covering a wide range of methods including SLOPE, PACS,
fused, clustered and classical LASSO as well as the related method of basis
pursuit. We consider a strong type of uniqueness that is relevant for
statistical problems. The uniqueness condition is geometric and involves how
the row span of the design matrix intersects the faces of the dual norm unit
ball, which for SLOPE is given by the signed permutahedron. Further
considerations based this condition also allow to derive results on sparsity
and clustering features. In particular, we define the notion of a SLOPE pattern
to describe both sparsity and clustering properties of this method and also
provide a geometric characterization of accessible SLOPE patterns.
