The Chebyshev Metric

# The Chebyshev Metric

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 a very important metric on $\mathbb{R}^n$ known as the Chebyshev metric.

 Definition: The Chebyshev Metric on $\mathbb{R}^n$ is the function $d : \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty)$ defined for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by $d(\mathbf{x}, \mathbf{y}) = \max_{1 \leq k \leq n} \mid x_k - y_k \mid$.

We must verify that the Chebyshev metric satisfies the three conditions to be a metric.

For the first condition we have that once again $\mid x_k - y_k \mid = \mid y_k - x_k \mid$ for all $k \in \{1, 2, ..., n \}$ so:

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

For the second condition, let $d(\mathbf{x}, \mathbf{y}) = 0$. Then $\max_{1 \leq k \leq n} \mid x_k - y_k \mid = 0$. But $\mid x_k - y_k \mid \geq 0$ for all $k \in \{1, 2, ..., n \}$ so $\mid x_k - y_k \mid = 0$ for each $k$ and $x_k - y_k = 0$ so $x_k = y_k$ for each $k$. Therefore $\mathbf{x} = \mathbf{y}$. Now let $\mathbf{x} = \mathbf{y}$. Then $\mid x_k - y_k \mid = 0$ for all $k \in \{1, 2, ..., n \}$ so:

(2)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \max_{1 \leq k \leq n} \mid x_k - y_k \mid = \max_{1 \leq k \leq n} 0 = 0 \end{align}

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

(3)
\begin{align} \quad d(\mathbf{x}, \mathbf{y}) = \max_{1 \leq k \leq n} \mid x_k - y_k \mid = \max_{1 \leq k \leq n} \mid x_k - z_k + z_k - y_k \mid \leq \max_{1 \leq k \leq n} [\mid x_k - z_k \mid + \mid z_k - y_k \mid] = \max_{1 \leq k \leq n} \mid x_k - z_k \mid + \max_{1 \leq k \leq 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.