```markdown
### Log-concave and Log-convex Functions

- A positive function \( f \) is **log-concave** if \( \log f \) is concave:
  \[
  f(\theta x + (1 - \theta) y) \geq f(x)^\theta f(y)^{1-\theta} \quad \text{for } 0 \leq \theta \leq 1
  \]

- \( f \) is **log-convex** if \( \log f \) is convex.

- **Examples**:
  - Powers: \( x^a \) on \( \mathbb{R}_{++} \) is log-convex for \( a \leq 0 \), log-concave for \( a \geq 0 \).
  - Many common probability densities are log-concave, e.g., normal distribution:
    \[
    f(x) = \frac{1}{\sqrt{(2\pi)^n \det \Sigma}} e^{-\frac{1}{2}(x-\bar{x})^T \Sigma^{-1} (x-\bar{x})}
    \]

- The cumulative Gaussian distribution function \( \Phi \) is log-concave:
  \[
  \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-u^2/2} \, du
  \]

- **Source**: A. d'Aspremont, Convex Optimization M2, Slide 28/67.
```
