Fifth Property of the Euclidean Metric
From Wikimization
(Difference between revisions)
Line 32: | Line 32: | ||
== References == | == References == | ||
- | * Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, | + | * Dattorro, [http://www.meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], Meboo, 2005 |
Revision as of 02:18, 10 January 2009
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, 2005