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: 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

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