```markdown
### Proof

- **Definition of Set \( \mathcal{A} \):**
  \[
  \mathcal{A} = \{(u, v, t) \mid \exists x \in D, \ f_i(x) \leq u_i, \ i = 1, \ldots, m, \ h_i(x) = v_i, \ i = 1, \ldots, p, \ f_0(x) \leq t\}
  \]
  This set represents the values taken by the constraint and objective functions.

- **Convexity of \( \mathcal{A} \):**
  If the problem is convex, \( \mathcal{A} \) is defined by a list of convex constraints, hence it is convex.

- **Definition of Set \( \mathcal{B} \):**
  \[
  \mathcal{B} = \{(0, 0, s) \in \mathbb{R}^m \times \mathbb{R}^p \times \mathbb{R} \mid s < p^\star\}
  \]

- **Non-Intersection of Sets \( \mathcal{A} \) and \( \mathcal{B} \):**
  The sets \( \mathcal{A} \) and \( \mathcal{B} \) do not intersect (otherwise \( p^\star \) could not be the optimal value of the problem).

- **First Step:**
  The hyperplane separating \( \mathcal{A} \) and \( \mathcal{B} \) defines a supporting hyperplane to \( \mathcal{A} \) at \( (0, p^\star) \).

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