```markdown
### Image Description

The image presents examples of different norms and their corresponding steepest descent directions:

1. **Euclidean Norm**: 
   - Formula: \(\Delta x_{sd} = -\nabla f(x)\)

2. **Quadratic Norm**: 
   - Formula: \(\|x\|_P = (x^T P x)^{1/2}\) (where \(P \in S_{++}^n\))
   - Steepest descent: \(\Delta x_{sd} = -P^{-1} \nabla f(x)\)

3. **\(\ell_1\)-Norm**: 
   - Formula: \(\Delta x_{sd} = -(\partial f(x)/\partial x_i) e_i\), where \(|\partial f(x)/\partial x_i| = \|\nabla f(x)\|_\infty\)

Additionally, the image includes diagrams illustrating unit balls and normalized steepest descent directions for a quadratic norm (represented by an ellipse) and the \(\ell_1\)-norm (represented by a diamond shape). Both diagrams show vectors \(-\nabla f(x)\) and \(\Delta x_{nsd}\) indicating the direction of steepest descent.
```
