Understanding Minkowski distance going to infinity

Question

$$d_p(x,y)=\sum^N_{n=1}|x_n-y_n|^p)^\frac{1}{p}, p = \infty$$

How can one intuitively understand the minkowski distance for $p=\infty$?

Answer 1

The Minkowski distance is defined for $p \geq 1$. So greater distances between components are given more weight (e.g. for $2^p > 4^p$). As $p \to \infty$ the bigger components are given more and more weight until only the largest component matters.

A bit more rigorously. Assume that $x \neq y$. We have $N$ components of $x,y$. Let's pick the biggest $|x_n - y_n|$ and denote it $M$ (which is non-zero since $x \neq y$). Then we have $\dfrac{|x_j - y_j|}{M} \leq 1 \; \forall j \in \{1, 2, \ldots, N\}$. Applying this to the given metric gives

\begin{align} d_p(x,y) &= \bigg(\sum_{n=1}^{N} |x_n - y_n|^p \bigg)^{1/p}\\ &= M \bigg(\sum_{n=1}^{N} \bigg(\dfrac{|x_n - y_n|}{M}\bigg)^p \bigg)^{1/p} \end{align}

Can you convince yourself $\lim_{p \to \infty} \bigg(\sum_{n=1}^{N} \bigg(\dfrac{|x_n - y_n|}{M}\bigg)^p \bigg)^{1/p} = 1$?