 Solution

Write $a_i^T$ for row $i$ of $A$, and define affine functionsThenThe pointwise maximum of convex functions is convex, so $f$ is a convex function. Moreover,so $f$ has Lipschitz continuity. The simpler bound $L\le \Vert c{\Vert }_2+M{\max }_i\Vert a_i{\Vert }_2$ is also valid.

Let $I(x)=\{i:{\ell }_i(x)=f(x)\}$ be the active set. The subdifferential isthe convex hull of all active slopes. In particular, choosing any active index gives the subgradientAt a tie, every convex combination of the tied slopes is also valid.