Minkowski.

G

G_{\mu \nu }\

G

G_{\mu \nu }\

{\mathcal {T}}

\psi

[

F\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} A.

][

D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.

] [

-i\hbar {\frac {\partial }{\partial x_{n}}}

.

\left|\psi ({\vec {r}},t)\right\rangle

The Minkowski distance or Minkowski metric is the distance function defined by a p-norm on a real coordinate space. It is a generalization of both the Euclidean distance ( $p=2$ ) and the Manhattan distance ( $p=1$ ). It is named after the mathematician Hermann Minkowski.

Definition

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The Minkowski distance of order $�$ (where $�$ is an integer) between two points $X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}$ is defined as: $D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.$

For $p\geq 1,$ the Minkowski distance is a metric as a result of the Minkowski inequality.^[1] When $p<1,$ the distance between $(0,0)$ and $(1,1)$ is $2^{1/p}>2,$ but the point $(0,1)$ is at a distance $1$ from both of these points. Since this violates the triangle inequality, for $p<1$ it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of $1/p.$ The resulting metric is also an F-norm.