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Define metrics $\rho$ and $d$ on the plane $\mathbb{R}^2$ as follows: for $x = (x_1, x_2)$ and $y = (y_1, y_2)$,

$$\rho(x, y) = |x_1 − y_1| + |x_2 − y_2|\\ d(x, y) = \max\{|x_1 − y_1|, |x_2 − y_2|\}$$

Draw accurate pictures in the $x$-$y$ plane of the unit neighborhoods about the origin $O = (0,0)$ in these two metrics; i.e., draw pictures of $N_1(O, \rho)$ and $N_1(O, d)$.

I am confused on the definition of the metrics. Why are there two values for $x$ and $y$? I'm really just struggling with grasping the definition of $\rho(x, y)$ and $d(x, y)$. Are they squares?

Qurultay
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Downieboy9
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2 Answers2

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You can think of a metric as a way of measuring distance between two objects in a way that is consistent with the "normal" way of measuring distance between points.

What this definition is saying is that, given the two points $(x_1, x_2)$ and $(y_1, y2)$, we can measure the "distance" between them in two ways. These ways to measure distance are described as the functions $\rho(x,y) = |x_1-y_1| + |x_2-y_2|$, and $d(x, y) = \max\{|x_1-y_1|,|x_2-y_2|\}$.

We have two values for $x$ and $y$ because $x$ and $y$ are points.

We define the "unit neighborhood of the origin" to be the set of all points such that the distance (as defined by our metric) between the point and the origin is $1$.

apnorton
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Points in the plane have two coordinates, $x=(x_1, x_2)$ is a point in $\mathbb R^2$ with coordinates $x_1$ and $x_2$.

Christoph
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