Fifth Property of the Euclidean Metric
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- | Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/ | + | Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]] |
namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math> | namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math> |
Revision as of 00:22, 1 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