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Suppose we have two vectors in $u, v \in \mathbb{R}^d$. For $p \geq 1$, the Minkowski distance between these vectors is defined as

$ \lVert u - v \rVert_p = \Bigl( \sum_{i=1}^d \lvert u_i - v_i \rvert^p \Bigr)^{1/p}. $

This family of distances includes the familiar Euclidean distance ($p = 2$), and the less familiar, but still physically meaningful Manhattan distance ($p = 1$).

When $1 < p < 2$, we can interpret the Minkowski distance as measuring (in a Euclidean sense) a path "between" the paths measured by the Manhattan and Euclidean distance, since

$ \lVert u - v \rVert_2 \leq \lVert u - v \rVert_p \leq \lVert u - v \rVert_1. $

Is there any physically meaningful or intuitive interpretation when $p > 2$?

Note: I am aware of this less specific question.

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Nick
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  • As $p$ gets bigger, the contributions of smaller terms get less and less, until only the maximum term is left. For instance, $\lVert(3,4)\rVert_1 = 7$, $\lVert(3,4)\rVert_2 = 5$, $\lVert(3,4)\rVert_3 \sim 4.5$, and $\lvert(3,4)\rVert_\infty = 4$. So raising $p$ "damps out" all the terms but the largest. – Nick Matteo Dec 10 '14 at 03:40
  • That's true, it becomes a chessboard distance. But it's not obvious to me how (if possible?) to relate a chessboard distance to a Euclidean path in physical space. – Nick Dec 10 '14 at 03:59

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