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 | ||
- | <math> | + | <math>d_{ij}=||x_i-x_j||^2 |
- | + | =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ | |
- | + | =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] | |
- | + | \left[\begin{array}{cc}x_i\\x_j\end{array}\right]</math> | |
- | \left[ | + | |
- | + | ||
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]] | 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]] |
Revision as of 23:56, 22 September 2016
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