Fifth Property of the Euclidean Metric

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relative angle inequality tetrahedron
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: \begin{array}{rl}d_{ij}
&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\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\neq 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\!\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

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.


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