I know that the Minkowski distance is defined as follows:
$$d(x,y)=\lim_{r\to\infty}\left(\sum_{k=1}^n|x_k-y_k|^r\right)^{1/r}$$
and I also read that the Chebyshev distance could be considered as a form of the Minkowski distance. I wanted to get a grasp by looking at the formula:
$$d(x,y)=\lim_{r\to\infty}\left(\sum_{k=1}^n|x_k-y_k|^r\right)^{1/r}=\underset{f}{\max}|x_{if}-x_{jf}|$$ but I cannot get the relationship. What I see is that if $r$ goes to $\infty$, then $\frac1r$ would be closer to $0$, so the (exp) would be closer to $1$, regardless if the absolute difference takes a big value from being raised to the power of $r$.
Anybody could help me to derive this equation?
Thanks