Fifth property of the Euclidean metric

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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 must satisfy the requirements imposed by any metric space:

• $\sqrt{d_{ij}}\geq0\,,~~i\neq j$ (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)

where $\sqrt{d_{ij}}$ is the Euclidean metric in $\mathbb{R}^n$.

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

$\begin{array}{cc} |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\quad\quad&{\rm(a)}\\ \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\quad\quad&{\rm(b)}\\ 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi\quad\quad&{\rm(c)} \end{array}$

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.