Exercise 8.2.1

Question

Decide which of the following are metrics on $X = \mathbf{R}^2$. For each, we let $x = (x_1,x_2)$ and $y = y_1, y_2$ be points in the plane.
\begin{enumerate}[label=(\alph*)] 
\item $d(x,y) \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$
\item $d(x,y) = \max\{|x_1 - y_1|, |x_2 - y_2|\}$
\item $d(x,y) = |x_1x_2 + y_1y_2|$
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item This is just the Euclidean distance between $x$ and $y$. The first two properties are obvious, while the third can be demonstrated with a little geometry.
\item It's fairly easy to see that the first two properties are met. To demonstrate the triangle inequality, note
$$d(x, z) + d(z, y) \geq \left|x_1 - z_1\right| + \left|z_1 - y_1\right| \geq \left|x_1 - y_1\right|$$
and
$$d(x, z) + d(z, y) \geq \left|x_2 - z_2\right| + \left|z_2 - y_2\right| \geq \left|x_2 - y_2\right|$$
therefore $d(x,z) + d(z,y)$ is greater than or equal to both possible values of $d(x,y)$.
\item Property (i) is not met; take $x = (0,1)$ and $y = (1,0)$.
 \end{enumerate}

Exercise 8.2.1

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Exercise 8.2.1

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