```markdown
## Examples on \( \mathbb{R} \)

### Convex:

- **Affine:** \( ax + b \) on \( \mathbb{R} \), for any \( a, b \in \mathbb{R} \)
- **Exponential:** \( e^{ax} \), for any \( a \in \mathbb{R} \)
- **Powers:** \( x^\alpha \) on \( \mathbb{R}_{++} \), for \( \alpha \geq 1 \) or \( \alpha \leq 0 \)
- **Powers of absolute value:** \( |x|^p \) on \( \mathbb{R} \), for \( p \geq 1 \)
- **Negative entropy:** \( x \log x \) on \( \mathbb{R}_{++} \)

### Concave:

- **Affine:** \( ax + b \) on \( \mathbb{R} \), for any \( a, b \in \mathbb{R} \)
- **Powers:** \( x^\alpha \) on \( \mathbb{R}_{++} \), for \( 0 \leq \alpha \leq 1 \)
- **Logarithm:** \( \log x \) on \( \mathbb{R}_{++} \)

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