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)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)For the third condition we have that by the triangle inequality:
(3)Therefore $(\mathbb{R}^n, d)$ is a metric space.