Let $X$ be a set equipped with a metric $d_x$, denoted by $\langle X,d_x\rangle$, and $Y$ equipped with a metric $d_y$, denoted by $\langle X,d_y\rangle$. Let $Z=X\times Y$.

Let $z_1=(x_1,y_1), \ z_2=(x_2,y_2), \ \forall z_1,z_2\in Z$, we define $d_p$ and $d_{\infty}$ by:

$(i) \ d_p(z_1,z_2)=(d_x(x_1,x_2)^p+d_y(y_1,y_2))^p)^{1/p}$, for $p \in \{1,2\}$ $(ii) \ d_\infty(z_1,z_2)=\max\{d_x(x_1,x_2),d_y(y_1,y_2)\}.$

Are $\langle Z,d_p\rangle$ and $\langle Z,d_\infty\rangle$ metric spaces?"

I have already shown that $d_p$ for $p=1$ is a distance function, the other two cases are now left (for $p=2$ and $d_{\infty}$).

I have already shown that all properties of a distance function are fullfilled except for the triangle inequality. How do I show this?


By Minkowski inequality, $d_p$ satisfies triangle inequality if $1\le p \le \infty$. It is easy that $d_p$ satisfies another conditions of definition of metric.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.