  The 1-norm was proven to be a good convex regularizer for the recovery of
sparse vectors from under-determined linear measurements. It has been shown
that with an appropriate measurement operator, a number of measurements of the
order of the sparsity of the signal (up to log factors) is sufficient for
stable and robust recovery. More recently, it has been shown that such recovery
results can be generalized to more general low-dimensional model sets and
(convex) regularizers. These results lead to the following question: to recover
a given low-dimensional model set from linear measurements, what is the "best"
convex regularizer? To approach this problem, we propose a general framework to
define several notions of "best regularizer" with respect to a low-dimensional
model. We show in the minimal case of sparse recovery in dimension 3 that the
1-norm is optimal for these notions. However, generalization of such results to
the n-dimensional case seems out of reach. To tackle this problem, we propose
looser notions of best regularizer and show that the 1-norm is optimal among
weighted 1-norms for sparse recovery within this framework.
