Fifth Property of the Euclidean Metric
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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 | 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 | ||
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==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== | ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== | ||
- | [[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]] | ||
Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>, | 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> , | for all <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> , |
Revision as of 03:39, 9 November 2007
For a list of points in Euclidean vector space, distance-square between points and is defined
Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [Dattorro, ch.5.2]
namely, for Euclidean metric in
- (nonnegativity)
- (self-distance)
- (symmetry)
- (triangle inequality)
Fifth property of the Euclidean metric (relative-angle inequality)
Augmenting the four fundamental Euclidean metric properties in , for all , , and for distinct points , the inequalities
where is the angle between vectors at vertex , must be satisfied at each point regardless of affine dimension.
References
- Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo, 2007