```markdown
### Perspective

The perspective of a function \( f : \mathbb{R}^n \rightarrow \mathbb{R} \) is the function \( g : \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R} \),

\[ 
g(x, t) = t f(x/t), \quad \text{dom } g = \{(x, t) \mid x/t \in \text{dom } f, \, t > 0\} 
\]

\( g \) is convex if \( f \) is convex.

#### Examples

- \( f(x) = x^T x \) is convex; hence \( g(x, t) = x^T x/t \) is convex for \( t > 0 \).
- Negative logarithm \( f(x) = -\log x \) is convex; hence relative entropy \( g(x, t) = t \log t - t \log x \) is convex on \( \mathbb{R}^2_{++} \).
- If \( f \) is convex, then

  \[
  g(x) = (c^T x + d) f \left( \frac{Ax + b}{c^T x + d} \right)
  \]

  is convex on \( \{ x \mid c^T x + d > 0, \, (Ax + b)/(c^T x + d) \in \text{dom } f \} \).

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