```markdown
### Minimization

If \( f(x, y) \) is convex in \( (x, y) \) and \( C \) is a convex set, then

\[
g(x) = \inf_{y \in C} f(x, y)
\]

is convex.

#### Examples

- \( f(x, y) = x^T A x + 2x^T B y + y^T C y \) with

  \[
  \begin{bmatrix}
  A & B \\
  B^T & C
  \end{bmatrix} \succeq 0, \quad C \succ 0
  \]

  Minimizing over \( y \) gives \( g(x) = \inf_y f(x, y) = x^T (A - BC^{-1}B^T)x \).

  \( g \) is convex, hence the Schur complement \( A - BC^{-1}B^T \succeq 0 \).

- Distance to a set: \( \text{dist}(x, S) = \inf_{y \in S} \|x - y\| \) is convex if \( S \) is convex.

*Source: A. d'Aspremont. Convex Optimization M2. Page 21/67.*
```