```markdown
### Properties of Log-Concave Functions

- A twice differentiable function \( f \) with a convex domain is log-concave if and only if:
  \[
  f(x) \nabla^2 f(x) \preceq \nabla f(x) \nabla f(x)^T
  \]
  for all \( x \in \text{dom } f \).

- The product of log-concave functions is log-concave.

- The sum of log-concave functions is not always log-concave.

- Integration: If \( f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R} \) is log-concave, then:
  \[
  g(x) = \int f(x, y) \, dy
  \]
  is log-concave (not easy to show).

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