```markdown
## Strong convexity and implications

- **Definition**: A function \( f \) is strongly convex on \( S \) if there exists an \( m > 0 \) such that:
  \[
  \nabla^2 f(x) \succeq mI \quad \text{for all } x \in S
  \]

### Implications

- **For \( x, y \in S \):**
  \[
  f(y) \geq f(x) + \nabla f(x)^T (y - x) + \frac{m}{2} \|x - y\|_2^2
  \]
  - Hence, \( S \) is bounded.

- **For \( p^* > -\infty \), and for \( x \in S \):**
  \[
  f(x) - p^* \leq \frac{1}{2m} \|\nabla f(x)\|_2^2
  \]
  - This is useful as a stopping criterion if \( m \) is known.

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