P-Product Metrics on Real Vector Space are Topologically Equivalent
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Theorem
For $n \in \N$, let $\R^n$ be an Euclidean space.
Let $p \in \R_{\ge 1}$.
Let $d_p$ be the $p$-product metric on $\R^n$.
Let $d_\infty$ be the Chebyshev distance on $\R^n$.
Then $d_p$ and $d_\infty$ are topologically equivalent.
Proof
Let $r, t \in \R_{\ge 1}$.
Without loss of generality, assume that $r \le t$.
For all $x, y \in \R^n$, we are going to show that:
- $\map {d_r} {x, y} \ge \map {d_\infty} {x, y} \ge n^{-1} \map {d_r} {x, y}$
Then we can demonstrate Lipschitz equivalence between all of these metrics, from which topological equivalence follows.
Let $d_r$ be the metric defined as:
- $\ds \map {d_r} {x, y} = \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^r}^{1/r}$
Inequalities for Chebyshev Distance
By definition of the Chebyshev distance on $\R^n$, we have:
- $\ds \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \size {x_i - y_i}$
where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.
Let $j$ be chosen so that:
- $\ds \size {x_j - y_j} = \max_{i \mathop = 1}^n \size {x_i - y_i}$
Then:
\(\ds \map {d_\infty} {x, y}\) | \(=\) | \(\ds \paren {\size {x_j - y_j}^p}^{1/p}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^p}^{1/p}\) | Power of Maximum is not Greater than Sum of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {d_p} {x, y}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \paren {n \size {x_j - y_j}^p}^{1/p}\) | Sum of $r$ Powers is not Greater than $r$ times Power of Maximum | |||||||||||
\(\ds \) | \(=\) | \(\ds n^{1/p} \size {d_\infty} {x, y}\) |
$\Box$
Inequality for General Case
We show that $\ds \map {d_r} {x, y} \ge \map {d_t} {x, y}$, which is equivalent to proving that:
- $\ds \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^r}^{1/r} \ge \paren {\sum_{i \mathop = 1}^n \size {x_i - y_i}^t}^{1/t}$
Let $\forall i \in \closedint 1 n: s_i = \size {x_i - y_i}$.
Suppose $s_k = 0$ for some $k \in \closedint 1 n$.
Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: Putting off the analysis of s_k being 0 until near the end of the proof should shorten it without losing any clarity. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Then the problem reduces to the equivalent one of showing that:
- $\ds \paren {\sum_{i \mathop = 1}^{n - 1} \size {x_i - y_i}^r}^{1/r} \ge \paren {\sum_{i \mathop = 1}^{n - 1} \size {x_i - y_i}^t}^{1/t}$
that is, of reducing the index by $1$.
Note that when $n = 1$, from simple algebra $\map {d_r} {x, y} = \map {d_t} {x, y}$.
So, let us start with the assumption that $\forall i \in \closedint 1 n: s_i > 0$.
Let $\ds u = \sum_{i \mathop = 1}^n \size {x_i - y_i}^r = \sum_{i \mathop = 1}^n s_i^r$, and $v = \dfrac 1 r$.
From Derivative of Function to Power of Function, $\map {D_r} {u^v} = v u^{v - 1} \map {D_r} u + u^v \ln u \map {D_r} v$.
Here:
- $\ds \map {D_r} u = \sum_{i \mathop = 1}^n s_i^r \ln s_i$ from Derivative of Exponential Function and Sum Rule for Derivatives
- $\map {D_r} v = - \dfrac 1 {r^2}$ from Power Rule for Derivatives
In the case where $r = 1$, we have:
- $\map {D_r} {u^v} = 0$
When $r > 1$, we have:
\(\ds \map {D_r} {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r} }\) | \(=\) | \(\ds \dfrac 1 r \paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / \paren {r - 1} } \paren {\sum_{i \mathop = 1}^n s_i^r \ln s_i} - \dfrac {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1/r} \map \ln {\sum_{i \mathop = 1}^n s_i^r} } {r^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r} } r \paren {\dfrac {\sum_{i \mathop = 1}^n s_i^r \ln s_i} {\sum_{i \mathop = 1}^n s_i^r} - \dfrac {\map \ln {\sum_{i \mathop = 1}^n s_i^r} } r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r} } r \paren {\dfrac {r \paren {\sum_{i \mathop = 1}^n s_i^r \ln s_i} - \paren {\sum_{i \mathop = 1}^n s_i^r} \map \ln {\sum_{i \mathop = 1}^n s_i^r} } {r \paren {\sum_{i \mathop = 1}^n s_i^r} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds K \paren {r \paren {\sum_{i \mathop = 1}^n s_i^r \ln s_i} - \paren {\sum_{i \mathop = 1}^n s_i^r} \map \ln {\sum_{i \mathop = 1}^n s_i^r} }\) | where $\ds K = \dfrac {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r} } {r^2 \paren {\sum_{i \mathop = 1}^n s_i^r} } > 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds K \paren {\sum_{i \mathop = 1}^n s_i^r \map \ln {s_i^r} - \paren {\sum_{i \mathop = 1}^n s_i^r} \map \ln {\sum_{i \mathop = 1}^n s_i^r} }\) | Logarithms of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds K \paren {\sum_{j \mathop = 1}^n \paren {s_j^r \paren {\map \ln {s_j^r} - \map \ln {\sum_{i \mathop = 1}^n s_i^r} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds K \paren {\sum_{j \mathop = 1}^n \paren {s_j^r \map \ln {\frac {s_j^r} {\sum_{i \mathop = 1}^n s_i^r} } } }\) |
where $K > 0$ because all of $s_i, r > 0$.
For the same reason, $\ds \dfrac{s_j^r} {\sum_{i \mathop = 1}^n s_i^r} < 1$ for all $j \in \set {1, \ldots, n}$.
From Logarithm of 1 is 0 and Logarithm is Strictly Increasing, their logarithms are therefore negative.
So for $r > 1$:
- $\ds \map {D_r} {\paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r} } < 0$
So, from Derivative of Monotone Function, it follows that (given the conditions on $r$ and $s_i$) $\ds \paren {\sum_{i \mathop = 1}^n s_i^r}^{1 / r}$ is decreasing.
As we assumed $r \le t$, we have $\map {d_r} {x, y} \ge \map {d_t} {x, y}$.
$\Box$
When we combine the inequalities, we have:
Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: In fact, this inequality (for real $r \ge 1$) can be established quite easily without the humongous calculation. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
- $\map {d_r} {x, y} \ge \map {d_\infty} {x, y} \ge n^{-1} \map {d_1} {x, y} \ge n^{-1} \map {d_r} {x, y}$
Therefore, $d_r$ and $d_\infty$ are Lipschitz equivalent for all $r \in \R_{\ge 1}$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Example $2.4.5$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 18$