Duality of block mixed norms
= Duality of block mixed norms
For a <linear map> $A$ between finite-dimensional real <inner product spaces>,
$$
\lambda\|Au\|_{q,1}=\sup_{\max_i\|p_i\|_{q'}\le\lambda}\langle A^*p,u\rangle.
$$
Thus the penalty is the <support function> of the <compact> <convex set> $A^*\{p:\max_i\|p_i\|_{q'}\le\lambda\}$. The <Moreau decomposition> makes its <proximal operator> the identity minus the <Euclidean projection onto a convex set>.