conjugate function for matrix mixed norm - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T05:03:29Z http://mathoverflow.net/feeds/question/92996 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92996/conjugate-function-for-matrix-mixed-norm conjugate function for matrix mixed norm Bernard 2012-04-03T12:46:10Z 2012-04-06T07:26:15Z <p>I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows:</p> <p>$\|\mathbf{y}\|_p^*=\max_{\mathbf{x}}\left(\mathbf{x}^T\mathbf{y}-\|\mathbf{x}\|_p\right)=\begin{cases}0~~~\|\mathbf{y}\|_q\leq 1 \\infty ~~~otherwise\end{cases}$ where $\frac{1}{p}+\frac{1}{q}=1$ for $p\geq 1$.</p> <p><strong>My question is:</strong></p> <p>Is there an equivalent conjugate function for the mixed matrix norm $\|\mathbf{A}\|_{p,q}$ defined for matrix $\mathbf{A}$?</p> <p>$\|\mathbf{A}\|_{p,q}=\left(\sum_i \|\mathbf{a}_i\|_p^q\right)^{1/q}$ where $\mathbf{a}_i$ is the $i^{\text{th}}$ column of matrix $\mathbf{A}$.</p> http://mathoverflow.net/questions/92996/conjugate-function-for-matrix-mixed-norm/93251#93251 Answer by S. Sra for conjugate function for matrix mixed norm S. Sra 2012-04-05T20:08:19Z 2012-04-05T20:08:19Z <p>Let <code>$p^*$</code> and <code>$q^*$</code> be the conjugate exponents. Some (slightly laborious) algebra shows that the dual-norm is <code>$\|A\|_{p^*,q^*}$</code>. The conjugate function is the indicator function for the (unit) dual-norm ball.</p>