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### Aufgabe 3 (Normierte Räume)

Given two points \( c \) and \( d \) in \(\mathbb{R}^2\):

- \( c = (0, 0) \)
- \( d = (4, 0) \)

We need to find the bisector of the two points using different norms.

#### a) Using the \( \ell_1 \) norm

The \( \ell_1 \) norm (Manhattan norm) of a vector \( x = (x_1, x_2) \) is defined as:
\[ \|x\|_1 = |x_1| + |x_2| \]

We need to find the set of points \( e = (x, y) \) such that the distance from \( e \) to \( c \) equals the distance from \( e \) to \( d \):
\[ \|e - c\|_1 = \|e - d\|_1 \]
\[ \|(x, y) - (0, 0)\|_1 = \|(x, y) - (4, 0)\|_1 \]
\[ |x| + |y| = |x - 4| + |y| \]

Considering different cases for \( x \) and \( x - 4 \):

1. If \( x \geq 4 \):
\[ x + |y| = x - 4 + |y| \]
\[ 0 = -4 \]
This case is not possible.

2. If \( x \leq 0 \):
\[ -x + |y| = 4 - x + |y| \]
\[ 0 = 4 \]
This case is also not possible.

3. If \( 0 \leq x \leq 4 \):
\[ x + |y| = 4 - x + |y| \]
\[ 2x = 4 \]
\[ x = 2 \]

So, the bisector is the vertical line \( x = 2 \).

#### b) Using the \( \ell_\infty \) norm

The \( \ell_\infty \) norm (maximum norm) of a vector \( x = (x_1, x_2) \) is defined as:
\[ \|x\|_\infty = \max(|x_1|, |x_2|) \]

We need to find the set of points \( e = (x, y) \) such that the distance from \( e \) to \( c \) equals the distance from \( e \) to \( d \):
\[ \|e - c\|_\infty = \|e - d\|_\infty \]
\[ \|(x, y) - (0, 0)\|_\infty = \|(x, y) - (4, 0)\|_\infty \]
\[ \max(|x|, |y|) = \max(|x - 4|, |y|) \]

Considering different cases for \( |x| \) and \( |x - 4| \):

1. If \( |x| = |y| \):
\[ |y| = \max(|x - 4|, |y|) \]
This is not possible as it does not provide new information.

2. If \( |x| > |y| \):
\[ |x| = \max(|x - 4|, |y|) = |x - 4| \]
This implies \( x = 2 \) or \( x = -2 \). However, \( x = -2 \) does not lie between 0 and 4, so \( x = 2 \).

3. If \( |x - 4| > |y| \):
\[ |x - 4| = \max(|x|, |y|) = |x| \]
This again implies \( x = 2 \).

So, the bisector in this case is also the vertical line \( x = 2 \).

### Summary
For both norms, the bisector of the points \( (0,0) \) and \( (4,0) \) is the vertical line \( x = 2 \).

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Would you like more details or have any questions about the solution?

Here are five additional questions you might ask:
1. How do other norms affect the bisector of two points?
2. Can you explain the concept of normed spaces in more detail?
3. What are the differences between the \( \ell_1 \), \( \ell_2 \), and \( \ell_\infty \) norms?
4. How can these norms be applied in higher dimensions?
5. What are some real-world applications of these mathematical concepts?

### Tip
Understanding the geometric interpretation of different norms can help visualize how distances and bisectors change in various normed spaces.

Explore how to find the bisector of two points \( (0, 0) \) and \( (4, 0) \) using the \( \ell_1 \) norm (Manhattan norm) and the \( \ell_\infty \) norm (maximum norm). Understand the geometric interpretations and calculations involved. Suitable for advanced mathematics enthusiasts and students.

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