```markdown
### Epigraph and Sublevel Set

#### α-sublevel set of \( f : \mathbb{R}^n \rightarrow \mathbb{R} \):

\[ C_\alpha = \{ x \in \text{dom } f \mid f(x) \leq \alpha \} \]

- Sublevel sets of convex functions are convex (the converse is false).

#### Epigraph of \( f : \mathbb{R}^n \rightarrow \mathbb{R} \):

\[ \text{epi } f = \{ (x, t) \in \mathbb{R}^{n+1} \mid x \in \text{dom } f, f(x) \leq t \} \]

- \( f \) is convex if and only if \(\text{epi } f\) is a convex set.

#### Illustration

- The image includes a graph illustrating the epigraph \(\text{epi } f\) as the shaded region above the curve \( f \).

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