```markdown
## The Conjugate Function

### Definition
The conjugate of a function \( f \) is defined as:

\[
f^*(y) = \sup_{x \in \text{dom } f} (y^T x - f(x))
\]

### Diagram
- The graph shows a plot of \( f(x) \) and a line representing \( xy \).
- A point is marked at \((0, -f^*(y))\).

### Properties
- \( f^* \) is convex (even if \( f \) is not).
- Used in regularization, duality results, etc.

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*Source: A. d'Aspremont, Convex Optimization M2, Slide 23/67.*
```