  We establish a result which states that regularizing an inverse problem with
the gauge of a convex set $C$ yields solutions which are linear combinations of
a few extreme points or elements of the extreme rays of $C$. These can be
understood as the \textit{atoms} of the regularizer. We then explicit that
general principle by using a few popular applications. In particular, we relate
it to the common wisdom that total gradient variation minimization favors the
reconstruction of piecewise constant images.
