```markdown
### Definition

- \( f: \mathbb{R}^n \rightarrow \mathbb{R} \) is convex if \(\text{dom } f\) is a convex set and

  \[
  f(\theta x + (1-\theta) y) \leq \theta f(x) + (1-\theta) f(y)
  \]

  for all \( x, y \in \text{dom } f \), \( 0 \leq \theta \leq 1 \).

- The diagram illustrates a convex function with points \((x, f(x))\) and \((y, f(y))\).

- \( f \) is concave if \(-f\) is convex.

- \( f \) is strictly convex if \(\text{dom } f\) is convex and

  \[
  f(\theta x + (1-\theta) y) < \theta f(x) + (1-\theta) f(y)
  \]

  for \( x, y \in \text{dom } f \), \( x \neq y \), \( 0 < \theta < 1 \).

---

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