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I have to show that $\delta$ is a metric with:

$$\delta(x,y):=\arctan(|x-y|)$$

The first two axioms are really straight forward, but I kinda struggle with showing

$$\arctan(|x-y|)\le\arctan(|x-z|)+\arctan(|y-z|)$$

My first try was (since the arctan is monotonic growing)

$$\arctan(|x-y|)\le\arctan(|x-z|+|y-z|)$$

But here Im stuck, I checked the graph just to make sure, the statement should be correct with saying:

$$\arctan(|x-z|+|y-z|)\le\arctan(|x-z|)+\arctan(|y-z|)$$

By plotting $\arctan(2x)$ and $2\arctan(x)$. The curve let me assume, that the statement:

$$\arctan(|x-z|+|y-z|)\le\arctan(|x-z|)+\arctan(|y-z|)$$

is indeed correct. But I cannot show it.

I tried going with taylor series, but I definitly cannot see how this is true:

$$\sum\limits_{k=0}^\infty (-1)^k\frac{(|x-z|+|y-z|)^{2k+1}}{2k+1}\le\sum\limits_{k=0}^\infty (-1)^k\frac{|x-z|^{2k+1}}{2k+1}+\sum\limits_{k=0}^\infty (-1)^k\frac{|y-z|^{2k+1}}{2k+1}$$

The addition theorems all are under conditions I cannot satisfy for my general statement. Or do I have to make cases?

Would be great if someone could give me a hint..

CoffeeArabica
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1 Answers1

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In general, let $f : [0, \infty) \to \mathbb{R}$ satisfy:

  • $f(0) = 0$,
  • $f$ is non-decreasing,
  • $f$ is concave.

Note that $\arctan(\cdot)$ satisfies this property. Then by concavity, for any $a, b \geq 0$ such that $a+b > 0$,

$$ f(a) = f\left( \frac{a}{a+b} (a+b) + \frac{b}{a+b} (0) \right) \geq \frac{a}{a+b}f(a+b) + \frac{b}{a+b}f(0) = \frac{a}{a+b}f(a+b). $$

By interchanging the role of $a$ and $b$, we also get

$$ f(b) \geq \frac{b}{a+b} f(a+b). $$

Then adding two inequality proves

$$ f(a+b) \leq f(a) + f(b). $$

Finally, since $f$ is increasing, for any $x, y, z \in \mathbb{R}$,

$$ f(|x-y|) \leq f(|x-z| + |z-y|) \leq f(|x-z|) + f(|y-z|). $$

Remark. More generally, if $d$ is a metric, then $f\circ d$ is also a metric. A standard application of this observation is the proof of the fact that any metrizable space admits a bounded metric that realizes its topology.

Sangchul Lee
  • 167,468