Fifth Property of the Euclidean Metric

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relative angle inequality tetrahedron

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

$LaTeX: d_{ij}=||x_i-x_j||^2 =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] \left[\begin{array}{cc}x_i\\x_j\end{array}\right]$

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 $LaTeX: \mathbb{R}^n$

$LaTeX: \sqrt{d_{ij}}\geq0\,,\;\;i\not= j$ (nonnegativity)
$LaTeX: \sqrt{d_{ij}}=0\;\Leftrightarrow\;x_i=x_j$ (self-distance)
$LaTeX: \sqrt{d_{ij}}=\sqrt{d_{ji}}$ (symmetry)
$LaTeX: \sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}\;,\;\;i\not=j\not=k$ (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 $LaTeX: i_{},j_{},\ell\not= k_{}\!\in\!\{1\,\ldots\,N\}\,,$ $LaTeX: i\!<\!j\!<\!\ell\,,$

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

Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo, 2005

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

@@ Line 1: / Line 1: @@
 [[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]]
-For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined
+For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell=1\,\ldots\,N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined
-<math>\begin{array}{rl}d_{ij}
+<math>d_{ij}=||x_i-x_j||^2
-\!\!&=\,\|x_i-_{}x_j\|^2
+=(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\
-~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\
+=\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right]
-&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right]
+\left[\begin{array}{cc}x_i\\x_j\end{array}\right]</math>
-\left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right]
-\end{array}</math>
 Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]]
 namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
-* <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''')'''
+* <math>\sqrt{d_{ij}}\geq0\,,\;\;i\not= 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''')'''
-* <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''')'''
+* <math>\sqrt{d_{ij}}=0\;\Leftrightarrow\;x_i=x_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;'''('''self-distance''')'''
 * <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''')'''
-* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math>  &nbsp;&nbsp;&nbsp;'''('''triangle inequality''')'''
+* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}\;,\;\;i\not=j\not=k</math>  &nbsp;&nbsp;&nbsp;'''('''triangle inequality''')'''
 ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''==
 Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
-&nbsp;for all &nbsp;<math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,
-&nbsp;<math>i\!<\!j\!<\!\ell</math> , &nbsp;and for &nbsp;<math>N\!\geq_{\!}4</math>&nbsp; distinct points &nbsp;<math>\{x_k\}</math> , &nbsp;the inequalities
+for all &nbsp;<math>i_{},j_{},\ell\not= k_{}\!\in\!\{1\,\ldots\,N\}\,,</math>
+&nbsp;<math>i\!<\!j\!<\!\ell\,,</math>
+and for &nbsp;<math>N\!\geq_{\!}4</math>&nbsp; distinct points &nbsp;<math>\,\{x_k\}\,,\,</math>
+the inequalities
 <math>\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}|\;\leq\;\theta_{ikj\!}\;\leq\;\theta_{ik\ell}+\,\theta_{\ell kj}\\
-\theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\\
+\theta_{ik\ell}+\,\theta_{\ell kj}+\,\theta_{ikj\!}\,\leq\,2\pi\\
-\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi
+\leq\theta_{ik\ell\,},\,\theta_{\ell kj\,},\,\theta_{ikj}\leq\pi
 \end{array}</math>
-where &nbsp;<math>\theta_{ikj}\!=_{}\!\theta_{jki}</math>&nbsp; is the angle between vectors at vertex &nbsp;<math>x_k</math> &nbsp;,  &nbsp;&nbsp;must be satisfied at each point &nbsp;<math>x_k</math>&nbsp; regardless of affine dimension.
+where &nbsp;<math>\theta_{ikj}\!=\theta_{jki}</math>&nbsp; is the angle between vectors at vertex <math>\,x_k\,</math>,
+must be satisfied at each point <math>\,x_k\,</math> regardless of affine dimension.
 == References ==
-* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007
+* Dattorro, [http://www.meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], Meboo, 2005

Fifth Property of the Euclidean Metric

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

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