Commit 55c16ce6 by Michael Kohlhase

### Minkowski distance

parent a899309b
 \begin{mhmodnl}[creators=miko]{Minkowski-distance}{de} \begin{definition} Ist $\realmorethan{p}1$ eine \mtrefii[realnumbers?real-number]{reelle}{Zahl}, so nennen wir die von der \mtrefi[pnorm?pnorm]{$\ell_p$-norm} auf $\ndim\RealNumbers{n}$ \mtrefii[norm-induced-metric?induced-metric]{induzierte}{Metrik} die \defi[name=Minkowski-distance]{Minkowskimetrik} $\MinkowskiDistOp{p}$. Es gilt $\fundefeq{p,x,y} {\MinkowskiDist{p}xy} {\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{\realminus{x_i,y_i}}}}{\frac1p}}$ f"ur $\defeq{x}{\ntupli{x}1n}$ und $\defeq{y}{\ntupli{y}1n}$. \end{definition} \begin{definition} Die durch die \mtrefi[pnorm?taxicab-norm]{Summennorm} \mtrefi[norm-induced-metric?induced-metric]{induzierte} \mtrefi[metric-space?metric]{Metrik} $\MinkowskiDistOp1$ hei"st \defi[name=Manhattan-distance]{Manhattan-Metrik}, \defi[name=Manhattan-distance]{Manhattan-Distanz}, \defii[name=Manhattan-distance]{Mannheimer}{Metrik}, \defi[name=Manhattan-distance]{Taxi-Metrik} oder \defi[name=Manhattan-distance]{Cityblock-Metrik}. \end{definition} \begin{definition} Die durch die \mtrefi[pnorm?pnorm]{$\ell_2$-Norm} \mtrefi[norm-induced-metric?induced-metric]{induzierte} \mtrefi[metric-space?metric]{Metrik} $\MinkowskiDistOp2$ hei"st \defii[name=Euclidean-distance]{Euklidsche}{Distanz}. \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{mhmodnl}[creators=miko]{Minkowski-distance}{en} \begin{definition} Let $\realmorethan{p}1$ be a \trefii[realnumbers]{real}{number}, then we call \atrefii[norm-induced-metric]{metric induced}{induced}{metric} on $\ndim\RealNumbers{n}$ by the \mtrefi[pnorm?pnorm]{$\ell_p$-norm} the \defii{Minkowski}{distance} $\MinkowskiDistOp{p}$. We have $\fundefeq{p,x,y} {\MinkowskiDist{p}xy} {\realpower[basebrack]{\Sumfromto{i}1n{\realabsval{\realminus{x_i,y_i}}}}{\frac1p}}$ where $\defeq{x}{\ntupli{x}1n}$ and $\defeq{y}{\ntupli{y}1n}$. \end{definition} \begin{definition} The \trefi[metric-space]{metric} $\MinkowskiDistOp1$ \atrefii[norm-induced-metric]{induced}{induced}{metric} by the \trefii[pnorm]{taxicab}{norm} is called the \defii{Manhattan}{distance}, the \defii[name=Manhattan-distance]{rectilinear}{distance}, \defii[name=Manhattan-distance]{snake}{distance}, or \defiii[name=Manhattan-distance]{city}{block}{distance}. \end{definition} \begin{definition} The \trefi[metric-space]{metric} $\MinkowskiDistOp2$ \atrefii[norm-induced-metric]{induced}{induced}{metric} by the \mtrefi[pnorm?pnorm]{$\ell_2$-norm} is called the \defii{Euclidean}{distance}. \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{modsig}[creators=miko]{Minkowski-distance} \gimport{pnorm} \gimport{norm-induced-metric} \symdef{MinkowskiDistOp}[1]{d_{#1}} \symdef{MinkowskiDist}[3]{\prefix{\MinkowskiDistOp{#1}}{#2,#3}} \symtest{MinkowskiDist}{\MinkowskiDist{p}xy} \end{modsig} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
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