```markdown
### Consequences of Integration Property

- **Convolution of Log-Concave Functions:**
  - If \( f \) and \( g \) are log-concave functions, their convolution \( f \ast g \) is also log-concave.
  - The convolution is defined as:
    \[
    (f \ast g)(x) = \int f(x - y)g(y) \, dy
    \]

- **Log-Concavity with Convex Sets:**
  - If \( C \subseteq \mathbb{R}^n \) is convex and \( y \) is a random variable with a log-concave probability density function (pdf), then:
    \[
    f(x) = \text{Prob}(x + y \in C)
    \]
  - This function \( f(x) \) is log-concave.
  - **Proof:**
    - Write \( f(x) \) as an integral of the product of log-concave functions:
      \[
      f(x) = \int g(x + y)p(y) \, dy
      \]
    - Where \( p \) is the pdf of \( y \) and:
      \[
      g(u) = 
      \begin{cases} 
      1 & \text{if } u \in C \\
      0 & \text{if } u \notin C 
      \end{cases}
      \]

#### Source
A. d'Aspremont. Convex Optimization M2.
```
