Fifth Property of the Euclidean Metric
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== References == | == References == | ||
- | * {{citation | first1 = Jon | last1 = Dattorro | year = 2007 | title = Convex Optimization & Euclidean Distance Geometry | url = http://www.convexoptimization.com | publisher = Meboo | isbn = 0976401304 }} | + | * {{citation | first1 = Jon | last1 = Dattorro | year = 2007 | title = Convex Optimization & Euclidean Distance Geometry | url = http://www.convexoptimization.com | publisher = Meboo | isbn = 0976401304 }} |
Revision as of 20:52, 14 October 2007
For a list of points in Euclidean vector space, distance-square between points and is defined
Euclidean distance must satisfy the requirements imposed by any metric space:
- (nonnegativity)
- (self-distance)
- (symmetry)
- (triangle inequality)
where is the Euclidean metric in .
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.