Minimum of two metrics is metric

Let X be a set and d, e two metrics on it. How can I show that the function f defined by f(x,y) = min(d(x,y), e(x,y)) is again a metric? This has been bothering me for the past few days as it seems intuitively true but always get stuck trying to prove the triangle inequality for f. Perhaps there is something obvious I'm missing.

If the minimums are the same metric the problem is easy, but showing d(x,y) + e(y,z) >= d(x,z) and the similar cases have left me stuck. My best attempt so far has been to attempt to find a contradiction by assuming the inequality is false:

f(x,y) + f(y,z) < f(x,z)

d(x,y) + e(y,z) < e(x,z) (a specific case in the contradiction, specifics not important)

d(x,y) + d(y,z) + e(y,z) < e(x,z) + d(y,z)

d(x,z) + e(y,z) < e(x,z) + d(y,z) (Using fact d is a metric)

d(x,z) + e(y,z) - d(y,z) < e(x,z), but e(y,z) < d(y,z) so d(x,z) should be smaller than e(x,z) which contradicts f(x,z) = e(x,z).

Now clearly this is an incorrect argument, but it's the best I have at this point. Any ideas?

Solution 1:

Counterexample : $$ d_1(x,y) = \sqrt{(x_1-y_1)^2 + \frac{1}{100} (x_2-y_2)^2} $$

$$ d_2(x,y) = \sqrt{\frac{1}{100}(x_1-y_1)^2 + (x_2-y_2)^2} $$

So let $x=(1,0),\ y=(0,1),\ z=(0,0)$ so that $$ d(x,y)>1 > d(x,z) + d(y,z)=\frac{1}{5}$$ where $d$ is a minimum.

Solution 2:

As people have pointed out, minimum of two metrics need not be a metric. The best way to construct examples and counterexamples of metric spaces is using the following method.

Let $X$ be any set, $(Y,d)$ be a metric space, and $f:X\to Y$ be an injective function. Define $d_f:X\times X\to \mathbb{R}$ as $d_f(x_1,x_2):=d(f(x_1),f(x_2))$. It is easy to verify that $d_f$ is a metric on $X$, let us call this as the metric induced by $f$.

Consider $X=\{a,b,c\}$, a three point set. Define $f,g:X\to \mathbb{R}$ as $f(a)=0,f(b)=1,f(c)=3$, and $g(a)=0,g(b)=2,g(c)=3$. Let $d_f$ and $d_g$ be the metrics on $X$ induced by the functions $f$ and $g$ respectively, where $\mathbb{R}$ is given the usual metric. We have \begin{align*} d_f(a,b)=1;\quad d_f(b,c)=2;\quad d_f(a,c)=3\\ d_g(a,b)=2;\quad d_g(b,c)=1;\quad d_g(a,c)=3. \end{align*} Hence, if we define $d:X\times X\to \mathbb{R}$ as $d(x,y):=\min\{d_f(x,y),d_g(x,y)\}$, then we have $$d(a,b)=1;\quad d(b,c)=1;\quad d(a,c)=3.$$ Clearly $$3=d(a,c)\nleq d(a,b)+d(b,c)=2,$$ i.e. the triangular inequality is not satisfied by $d$, and hence it can not be a metric.