Fifth Property of the Euclidean Metric
From Wikimization
(Difference between revisions)
Line 13: | Line 13: | ||
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> | ||
* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> '''('''nonnegativity''')''' | * <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> '''('''nonnegativity''')''' | ||
- | * <math>\sqrt{d_{ij}}=0 | + | * <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math> '''('''self-distance''')''' |
* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> '''('''symmetry''')''' | * <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> '''('''symmetry''')''' | ||
* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> '''('''triangle inequality''')''' | * <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> '''('''triangle inequality''')''' |
Revision as of 02:23, 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