Fifth Property of the Euclidean Metric

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namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
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* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> '''('''nonnegativity''')'''
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* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''nonnegativity''')'''
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* <math>\sqrt{d_{ij}}=0\,,~~i=j</math> '''('''self-distance''')'''
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* <math>\sqrt{d_{ij}}=0\,,~~i=j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''self-distance''')'''
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* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> '''('''symmetry''')'''
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* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''symmetry''')'''
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* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> '''('''triangle inequality''')'''
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* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> &nbsp;&nbsp;&nbsp;'''('''triangle inequality''')'''
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==Fifth property of the Euclidean metric ==
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==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==
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'''('''Relative-angle inequality.''')'''
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Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
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<math>\begin{array}{cc}
<math>\begin{array}{cc}
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|\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\quad\quad&{\rm(a)}\\
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|\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\\
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\theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\quad\quad&{\rm(b)}\\
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\theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\\
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0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi\quad\quad&{\rm(c)}
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0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi
\end{array}</math>
\end{array}</math>

Revision as of 01:09, 31 October 2007

For a list of points LaTeX: \{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\} in Euclidean vector space, distance-square between points LaTeX: x_i and LaTeX: x_j is defined

LaTeX: \begin{array}{rl}d_{ij} \!\!&=\,\|x_i-_{}x_j\|^2 ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right] \left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right] \end{array}

Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [Dattorro, ch.5.2]

namely, for Euclidean metric LaTeX: \sqrt{d_{ij}} in LaTeX: \mathbb{R}^n

  • LaTeX: \sqrt{d_{ij}}\geq0\,,~~i\neq j                                       (nonnegativity)
  • LaTeX: \sqrt{d_{ij}}=0\,,~~i=j                                       (self-distance)
  • LaTeX: \sqrt{d_{ij}}=\sqrt{d_{ji}}                                                  (symmetry)
  • LaTeX: \sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k    (triangle inequality)


Fifth property of the Euclidean metric (relative-angle inequality)

Augmenting the four fundamental Euclidean metric properties in LaTeX: \mathbb{R}^n, for all LaTeX: i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\} , LaTeX: i\!<\!j\!<\!\ell , and for LaTeX: N\!\geq_{\!}4 distinct points LaTeX: \{x_k\} , the inequalities

LaTeX: \begin{array}{cc} |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\\ \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\\ 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi \end{array}

where LaTeX: \theta_{ikj}\!=_{}\!\theta_{jki} is the angle between vectors at vertex LaTeX: x_k , must be satisfied at each point LaTeX: x_k regardless of affine dimension.

References

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Fifth_Property_of_the_Euclidean_Metric"
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