View source for Fifth property of the Euclidean metric

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}
\!\!&=\,\|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}</math>

[[Euclidean distance]] must satisfy the requirements imposed by any metric space: 

{{harvtxt|Dattorro|2007, ch.5.2}}
* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math>  '''('''nonnegativity''')'''
* <math>\sqrt{d_{ij}}=0\,,~~i=j</math>  '''('''self-distance''')'''
* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math>  '''('''symmetry''')'''
* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math>  '''('''triangle inequality''')'''

where <math>\sqrt{d_{ij}}</math> is the Euclidean metric in <math>\mathbb{R}^n</math>.

==Fifth property of the Euclidean metric ==
'''('''Relative-angle inequality.''')'''

Augmenting the four fundamental [[Euclidean metric]] properties in <math>\mathbb{R}^n</math>,
for all <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}_{\,}</math>,
<math>i\!<\!j\!<\!\ell\,\,</math>, and for <math>N\!\geq_{\!}4</math> distinct points <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}\quad\quad&{\rm(a)}\\
\theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\quad\quad&{\rm(b)}\\
0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi\quad\quad&{\rm(c)}
\end{array}</math>

where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> 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 ==
* {{citation | first1 = Jon | last1 = Dattorro | year = 2007 | title = Convex Optimization & Euclidean Distance Geometry | url = http://www.convexoptimization.com | publisher = Meboo | isbn = 0976401304 }}.