[Math] Showing that product of two metrics is not always a metric

metric-spaces

Let $d_1$ and $d_2$ be two metrics on space $X$. Is $d_1\cdot d_2$ metric on space $X$?

I know that is satisfies first three properties of metric. How do I show that triangle inequality holds or does not hold (an example) for this?

Best Answer

Consider the real line $\mathbb R$ with metrics $d_1(x,y)=|x-y|$ and $d_2(x,y)=\min\{1,d_1(x,y)\}$. The triangle inequality should hold for $d_1\cdot d_2$ for all points $x,y,z\in\mathbb R$ for it to be a metric.

Consider the points $x=1,y=2,z=2.5$ we have that -

$d_1(1,2.5)\cdot d_2(1,2.5)=|1-2.5|\cdot\min\{1,|1-2.5|\}=1.5\cdot 1=1.5$

$d_1(1,2)\cdot d_2(1,2)+d_1(2,2.5)\cdot d_2(2,2.5)=1\cdot1+0.5\cdot0.5=1.25<1.5$

Thus triangle inequality doesn't hold.

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