```markdown
### Pointwise Supremum

If \( f(x, y) \) is convex in \( x \) for each \( y \in A \), then 

\[ 
g(x) = \sup_{y \in A} f(x, y) 
\]

is convex.

#### Examples

- **Support function of a set \( C \):** 

  \[
  S_C(x) = \sup_{y \in C} y^T x 
  \]

  is convex.

- **Distance to farthest point in a set \( C \):**

  \[
  f(x) = \sup_{y \in C} \|x - y\|
  \]

- **Maximum eigenvalue of symmetric matrix:** For \( X \in S^n \),

  \[
  \lambda_{\max}(X) = \sup_{\|y\|_2 = 1} y^T X y
  \]

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*A. d’Aspremont. Convex Optimization M2.*  
Page 18/67
```