```markdown
### Chebyshev Center of a Polyhedron

The slide describes the concept of the Chebyshev center of a polyhedron. 

- **Polyhedron Definition**: 
  \[
  \mathcal{P} = \{ x \mid a_i^T x \leq b_i, \, i = 1, \ldots, m \}
  \]
  The Chebyshev center is the center of the largest inscribed ball within this polyhedron.

- **Inscribed Ball**: 
  \[
  \mathcal{B} = \{ x_c + u \mid \|u\|_2 \leq r \}
  \]

- **Condition for Containment**:
  \[
  a_i^T x \leq b_i \text{ for all } x \in \mathcal{B} \text{ if and only if } \sup \{ a_i^T(x_c + u) \mid \|u\|_2 \leq r \} = a_i^T x_c + r \|a_i\|_2 \leq b_i
  \]

- **Optimization Problem**:
  \[
  \text{maximize } r \quad \text{subject to } \quad a_i^T x_c + r \|a_i\|_2 \leq b_i, \, i = 1, \ldots, m
  \]

The diagram illustrates the largest inscribed circle within a polygon, with the center labeled as \( x_{\text{cheb}} \).

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