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First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. Summary: The authors re-explain regularization in optimization problems as a constraint of the type the parameters ${\bf w}$ must belong to the convex set $O$ where the convex set O is obtained as the convex hull of all the points of the form $g.v$ where $v$ is some fix vector, $g$ an element from a group and $.$ is a (linear) group action of element $g$ on vector $v$. More concretely, their main contributions are as follows. For example, the ball associated to the L1 norm can be explained as the convex hull of the points obtained by flipping the sign and permuting the components of the vector $(1,0,0,..,0)$; (B) they show that given a seed $v$ and a group action associated to a group $G$, the notion of $w$ is a member of the convex set $O_G(v)$ can be seen as $v$ is smaller than $w$ under a pre-order; (C) they show that if $-v$ belongs to convex set $O$ then $O$ can be seen as the ball of an atomic norm (as defined in Chandra et al.); (D) they show that the L1-sorted norm equals the dual of the norm associated to the signed-pertumation orbitope; (E) they show how to reinterpret the main steps of conditional and projected gradient algorithms in the language of orbitopes and give a procedure to compute projections onto orbitopes. Quality: There are no technical mistakes in the paper.