The mixed $\ell_1/\ell_2$ norm $\Omega_{12} $ is defined as $\Omega_{12}(x) = \sum_g ||x_g||_2$ where $x_g$ are disjoint subsets of the elements of the vector $x$. This is used in machine learning applications as a regularizer that induces sparsity in groups of variables. What is the dual norm of $\Omega_{12}$? The dual of the $\ell_1$ norm is the $\ell_\infty$ norm, and the $\ell_2$ norm is self-dual. Is the dual of $\Omega_{12}$ just $\Omega_{\infty 2}$?
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2Yes (see the second part, starting with "More generally") – Daniel Fischer Feb 10 '14 at 23:02