# Subdifferential of a finite dimensional function.

I want to compute the subgradients of the absolute value function in $\mathbb{R}^n$. How do I do this?

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By absolute value, you mean $|x|= \left( \sum_i x_i^2 \right)^{1/2}$? – Seirios Nov 3 '12 at 11:45

$\phi=||\cdot||$ is differentiable on $\mathbb{R}^n \backslash \{0\}$, so $\partial \phi(x)=\{\phi'(x) \}$ (you can show that $\phi'(x)= \frac{1}{|x|} \langle x, \cdot \rangle$). Then, $\partial \phi(0)= \{ \zeta \in (\mathbb{R}^n)^* \ | \ \forall y \in \mathbb{R}^n, |y| \geq \langle \zeta, y \rangle \}$ is the unit closed ball in $(\mathbb{R}^n)^*$.