```markdown
### Positive Weighted Sum & Composition with Affine Function

#### Nonnegative Multiple
- \( \alpha f \) is convex if \( f \) is convex, \( \alpha \geq 0 \).

#### Sum
- \( f_1 + f_2 \) is convex if \( f_1, f_2 \) are convex (extends to infinite sums, integrals).

#### Composition with Affine Function
- \( f(Ax + b) \) is convex if \( f \) is convex.

#### Examples

- **Log Barrier for Linear Inequalities**
  \[
  f(x) = -\sum_{i=1}^{m} \log(b_i - a_i^T x), \quad \text{dom } f = \{ x \mid a_i^T x < b_i, i = 1, \ldots, m \}
  \]

- **(Any) Norm of Affine Function**
  \[
  f(x) = \| Ax + b \|
  \]

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