Some Metrics Defined on Euclidean Space

# Some Metrics Defined on Euclidean Space

Recall from the Metric Spaces page that if $M$ is a nonempty set then a function $d : M \times M \to [0, \infty)$ is called a metric if for all $x, y, z \in M$ we have that the following three properties hold:

• $d(x, y) = d(y, x)$.
• $d(x, y) = 0$ if and only if $x = y$.
• $d(x, y) \leq d(x, z) + d(z, y)$.

Furthermore, the set $M$ with the metric $d$, denoted $(M, d)$ is called a metric space.

We will now look at some other metrics defined on the Euclidean space $\mathbb{R}^n$ specifically.

The first type of metric $d : \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty)$ is defined for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n), \mathbf{z} = (z_1, z_2, ..., z_n) \in \mathbb{R}^n$ by:

(1)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid \end{align}

Let's verify that $d$ is indeed a metric.

For the first condition we have that for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ that since $\mid x_k - y_k \mid = \mid y_k - x_k \mid$ that then:

(2)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid = \sum_{k=1}^{n} \mid y_k - x_k \mid = d(\mathbf{y}, \mathbf{x}) \end{align}

For the second condition, suppose that $d(\mathbf{x}, \mathbf{y}) = 0$. Then:

(3)
\begin{align} \quad \sum_{k=1}^{n} \mid x_k - y_k \mid = \mid x_1 - y_1 \mid + \mid x_2 - y_2 \mid + ... + \mid x_n - y_n \mid = 0 \end{align}

We have that $\mid y_k - x_k \mid \geq 0$ for all $k \in \{1, 2, ..., n \}$ so for the sum above to equal to $0$, we must have that $\mid y_k - x_k \mid = 0$ for each $k$, so $y_k - x_k = 0$ and $y_k = x_k$ for each $k$. Hence $\mathbf{x} = \mathbf{y}$. Now suppose that $\mathbf{x} = \mathbf{y}$. Then $x_k = y_k$ for each $k \in \{ 1, 2, ..., n \}$ so $\mid x_k - y_k \mid = 0$ for each $k$ and:

(4)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid = \sum_{k=1}^{n} 0 = 0 \end{align}

For the third condition we have by the triangle inequality that:

(5)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n} \mid x_k - y_k \mid = \sum_{k=1}^{n} \mid x_k - z_k + z_k - y_k \mid \leq \sum_{k=1}^{n} [\mid x_k - z_k \mid + \mid z_k - y_k \mid] = \sum_{k=1}^{n} \mid x_k - z_k \mid + \sum_{k=1}^{n} \mid z_k - y_k \mid = d(\mathbf{x}, \mathbf{z}) + d(\mathbf{z}, \mathbf{y}) \end{align}

Therefore $(\mathbb{R}^n, d)$ is a metric space.