91Ó°ÊÓ

The L1 Distance, often called the "Taxicab" or "Manhattan" distance, gets its name from the way a taxi would travel on a grid of city streets. It measures the distance between two points by adding up the absolute differences of their corresponding coordinates.
For any two vectors, \(x = (x_1, \ldots, x_k)\) and \(y = (y_1, \ldots, y_k)\) in \(\mathbb{R}^k\), the L1 distance is represented as:

• \(|x-y|_1 = \sum_{i=1}^{k} |x_i - y_i|\).

This formula sums up how far each component of \(x\) is from \(y\) and adds the absolute values of these differences.

To confirm that this is indeed a true distance metric, we must ensure it meets certain criteria:

• **Non-negativity:** Since absolute values are non-negative, the total sum is always \(\geq 0\).
• **Identity of Indiscernibles:** If \(|x-y|_1 = 0\), then all \(|x_i - y_i| = 0\) indicating \(x = y\).
• **Symmetry:** The path between two points is the same in reverse, thus \(|x-y|_1 = |y-x|_1\).
• **Triangle Inequality:** For any third vector \(z\), the sum of distances from \(x\) to \(z\) plus \(z\) to \(y\) covers at least as much as \(x\) to \(y\).

Understanding these properties ties the technical side of the L1 Distance to practical navigation or measurement tasks.

The L∞ Distance is a bit different from its L1 counterpart. It emphasizes the maximum distance of all dimensions, rather than the sum. This metric captures the most significant individual gap between points, measuring the farthest dimension.
For vectors \(x = (x_1, \ldots, x_k)\) and \(y = (y_1, \ldots, y_k)\), the L∞ distance is defined as:

• \(|x-y|_{\infty} = \max(|x_1 - y_1|, \ldots, |x_k - y_k|)\).

Here, we only consider the largest difference between corresponding components of \(x\) and \(y\).

For \(|x-y|_{\infty}\) to be a valid distance measure, it must fulfill these criteria:

• **Non-negativity:** Since we're looking at maximum absolute differences, \(|x-y|_{\infty} \geq 0\).
• **Identity of Indiscernibles:** If \(|x-y|_{\infty} = 0\), all component distances are zero, making \(x = y\).
• **Symmetry:** Flipping the role of \(x\) and \(y\) doesn't change the maximum distance: \(|x-y|_{\infty} = |y-x|_{\infty}\).
• **Triangle Inequality:** For an intermediate point \(z\), the magnitude of maximum individual shifts respects the triangle inequality rule.
  

This measure gives insight into the largest potential disruptor in any dimension between two points.

The Triangle Inequality is a fundamental concept in metric spaces. It helps ensure that the shortest distance between two points is a direct path and not via a third point.
In mathematical terms, for any points \(x, y,\) and \(z\) in a metric space, it states:

• \(|x-y| \leq |x-z| + |z-y|\).

This principle guarantees that the total of the two legs of a triangle will never be less than the direct distance across. The inequality holds true for both L1 and L∞ distances.

For L1 Distance, each dimension holds to the absolute value rules:

• \(|x_i - y_i| \leq |x_i - z_i| + |z_i - y_i|\).

When summed across all components, this satisfies the inequality for the entire distance.
For L∞ Distance, although we only consider the maximum component, the principle is similar:

• \(|x-y|_{\infty} \leq |x-z|_{\infty} + |z-y|_{\infty}\).

The "maximum" nature allows respect for this inequality.

The Triangle Inequality assures the consistency and logic of a distance metric, affirming that detours are never shorter than the direct way.