```markdown
### Proof

**Second step:** Slater’s constraint qualification is used to establish that the hyperplane must be nonvertical, i.e., \(\mu > 0\).

**By contradiction,** assume that \(\mu = 0\). From (3), we conclude that for all \(x \in \mathcal{D}\),

\[
\sum_{i=1}^{m} \tilde{\lambda}_i f_i(x) + \tilde{\nu}^T(Ax - b) \geq 0. \quad (4)
\]

- Applying this to the point \(\tilde{x}\) that satisfies the Slater condition, we have

\[
\sum_{i=1}^{m} \tilde{\lambda}_i f_i(\tilde{x}) \geq 0.
\]

- Since \(f_i(\tilde{x}) < 0\) and \(\tilde{\lambda}_i \geq 0\), we conclude that \(\tilde{\lambda} = 0\).

---

*A. d’Aspremont. Convex Optimization M2.*  
*48/49*
```
