```markdown
## Generalized Logarithm for Proper Cone

### Definition
The function \( \psi : \mathbb{R}^q \to \mathbb{R} \) is a generalized logarithm for a proper cone \( K \subseteq \mathbb{R}^q \) if:

- **Domain Condition**: \( \text{dom } \psi = \text{int } K \) and \( \nabla^2 \psi(y) \prec 0 \) for \( y \succ_K 0 \)
- **Scaling Property**: \( \psi(sy) = \psi(y) + \theta \log s \) for \( y \succ_K 0, s > 0 \) (\( \theta \) is the degree of \( \psi \))

### Examples

- **Nonnegative Orthant** \( K = \mathbb{R}^n_+ \):
  \[
  \psi(y) = \sum_{i=1}^n \log y_i, \quad \text{with degree } \theta = n
  \]

- **Positive Semidefinite Cone** \( K = \mathbb{S}^n_+ \):
  \[
  \psi(Y) = \log \det Y \quad (\theta = n)
  \]

- **Second-Order Cone** \( K = \{ y \in \mathbb{R}^{n+1} \mid (y_1^2 + \cdots + y_n^2)^{1/2} \leq y_{n+1} \} \):
  \[
  \psi(y) = \log(y_{n+1}^2 - y_1^2 - \cdots - y_n^2) \quad (\theta = 2)
  \]

*Slide reference: A. d’Aspremont. Convex Optimization M2.*
```
