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What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for a $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$

$\|\mathbf{x}\| = \displaystyle{ \sum_{i=0}^{n} \|\mathbf{A}_i \cdot \mathbf{x} \|_2}$.

How would you then find

$\|\mathbf{y}\|_* = \underset{\mathbf{x}}{\mathrm{max}} \left\{ |\mathbf{y}^T \cdot \mathbf{x}| \;\; \mathrm{s.t.} \;\; \|\mathbf{x}\| \leq 1\right\}$?

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