```markdown
## Equality Constrained Norm Minimization

### Objective
- **Minimize**: \( \|x\| \)
- **Subject to**: \( Ax = b \)

### Dual Function
\[ 
g(\nu) = \inf_x (\|x\| - \nu^T Ax + b^T \nu) = 
\begin{cases} 
b^T \nu & \|A^T \nu\|_* \leq 1 \\
-\infty & \text{otherwise}
\end{cases} 
\]

- **Where**: \( \|v\|_* = \sup_{\|u\| \leq 1} u^T v \) is the dual norm of \( \|\cdot\| \).

### Proof
- Follows from \( \inf_x (\|x\| - y^T x) = 0 \) if \( \|y\|_* \leq 1 \), \(-\infty\) otherwise.

- **Case Analysis**:
  - If \( \|y\|_* \leq 1 \), then \( \|x\| - y^T x \geq 0 \) for all \( x \), with equality if \( x = 0 \).
  - If \( \|y\|_* > 1 \), choose \( x = tu \) where \( \|u\| \leq 1 \), \( u^T y = \|y\|_* > 1 \):
    \[
    \|x\| - y^T x = t(\|u\| - \|y\|_*) \to -\infty \quad \text{as } t \to \infty
    \]

### Lower Bound Property
\[ 
p^\star \geq b^T \nu \quad \text{if } \|A^T \nu\|_* \leq 1 
\]

---
*Source: A. d'Aspremont. Convex Optimization M2.*
```
