Fifth Property of the Euclidean Metric
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\!\!&=\,\|x_i-_{}x_j\|^2 | \!\!&=\,\|x_i-_{}x_j\|^2 | ||
~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ | ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ | ||
- | &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{ | + | &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right] |
- | \left[\!\!\begin{array}{ | + | \left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right] |
\end{array}</math> | \end{array}</math> | ||
Revision as of 14:07, 19 November 2011
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