  This paper deals with the non-uniqueness of the solutions of an
analysis-Lasso regularization. Most of previous works in this area is concerned
with the case where the solution set is a singleton, or to derive guarantees to
enforce uniqueness. Our main contribution consists in providing a geometrical
interpretation of a solution with a maximal D-support, namely the fact that
such a solution lives in the relative interior of the solution set. With this
result in hand, we also provide a way to exhibit a maximal solution using a
primal-dual interior point algorithm.
