Fifth Property of the Euclidean Metric

From Wikimization

Jump to: navigation, search
relative angle inequality tetrahedron
relative angle inequality tetrahedron

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

\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 \sqrt{d_{ij}} in \mathbb{R}^n

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


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

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

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

[edit] References

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