d p ( x , y ) = ∑ n = 1 N | x n − y n | p ) 1 p , p = ∞How can one intuitively understand the minkowski distance for p = ∞?

Question

d    p    (  x  ,  y  )  =      ∑          n      =      1        N        |        x    n    −      y    n              |        p        )          1      p        ,  p  =  ∞How can one intuitively understand the minkowski distance for   p  =  ∞?

Hugh Petty · Accepted Answer

The Minkowski distance is defined for   p  ≥  1. So greater distances between components are given more weight (e.g. for       2    p    &amp;gt;      4    p  ). As   p  →  ∞ the bigger components are given more and more weight until only the largest component matters.A bit more rigorously. Assume that   x  ≠  y. We have N components of x,y. Let&#039;s pick the biggest       |        x    n    −      y    n        |   and denote it M (which is non-zero since   x  ≠  y). Then we have                               |                          x          j                −                  y          j                          |                    M        ≤  1    ∀  j  ∈  {  1  ,  2  ,  …  ,  N  }. Applying this to the given metric gives                              d          p                (        x        ,        y        )                            =                              (                                    ∑                      n            =            1                                N                                    |                          x          n                −                  y          n                                      |                    p                                                    )                                            1                          /                        p                                                            =        M                              (                                    ∑                      n            =            1                                N                                                (                                                                              |                                            x                n                            −                              y                n                                            |                                      M                                                              )                                p                                                    )                                            1                          /                        p

d_p(x,y)=sum_(n=1)^N |x_n−y_n|^p)^(1/p),p=oo How can one intuitively understand the minkowski distance for p=oo?

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