```markdown
### Second-order conditions

- **Twice Differentiable Function**:  
  \( f \) is twice differentiable if \( \text{dom } f \) is open and the Hessian \( \nabla^2 f(x) \in S^n \),
  \[
  (\nabla^2 f(x))_{ij} = \frac{\partial^2 f(x)}{\partial x_i \partial x_j}, \quad i, j = 1, \ldots, n,
  \]
  exists at each \( x \in \text{dom } f \).

- **2nd-order conditions**: For twice differentiable \( f \) with convex domain:
  - \( f \) is convex if and only if 
    \[
    \nabla^2 f(x) \succeq 0 \quad \text{for all } x \in \text{dom } f
    \]
  - If \( \nabla^2 f(x) \succ 0 \) for all \( x \in \text{dom } f \), then \( f \) is strictly convex.

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