```markdown
The image contains mathematical content discussing the convexity of the log-sum-exp function and the concavity of the geometric mean. 

### Log-Sum-Exp Function
- **Function Definition**: \( f(x) = \log \sum_{k=1}^{n} \exp x_k \)
- **Convexity Proof**:
  - The Hessian \( \nabla^2 f(x) \) is given by:
    \[
    \nabla^2 f(x) = \frac{1}{\mathbf{1}^T z} \text{diag}(z) - \frac{1}{(\mathbf{1}^T z)^2} zz^T
    \]
    where \( z_k = \exp x_k \).
  - To show \( \nabla^2 f(x) \succeq 0 \), verify:
    \[
    v^T \nabla^2 f(x) v = \frac{(\sum_k z_k v_k^2)(\sum_k z_k) - (\sum_k v_k z_k)^2}{(\sum_k z_k)^2} \geq 0
    \]
  - This inequality holds due to the Cauchy-Schwarz inequality:
    \[
    (\sum_k v_k z_k)^2 \leq (\sum_k z_k v_k^2)(\sum_k z_k)
    \]

### Geometric Mean
- **Function Definition**: \( f(x) = \left( \prod_{k=1}^{n} x_k \right)^{1/n} \) on \( \mathbb{R}_{++}^n \)
- **Concavity Proof**: A similar proof to the log-sum-exp is suggested.

The content is attributed to A. d’Aspremont, from a lecture on Convex Optimization.
```
