Gauge Distances and Median Hyperplanes

By F. Plastria, Emilio Carrizosa. 2001

49 citations recorded by the source corpus.

A median hyperplane in d-dimensional space minimizes the weighted sum of the distances from a finite set of points to it. When the distances from these points are measured by possibly different gauges, we prove the existence of a median hyperplane passing through at least one of the points. When all the gauges are equal, some median hyperplane will pass through at least d-1 points, this number being increased to d when the gauge is symmetric, i.e. the gauge is a norm.Whereas some of these results have been obtained previously by different methods, we show that they all derive from a simple formula for the distance of a point to a hyperplane as measured by an arbitrary gauge.

Subfields: Computer Graphics and Computer-Aided Design, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5c3466e3d3d65019c33cf52caecac91c54ad8a54

Citations in the corpus

• Supervised classification and mathematical optimization (2013, unknown): “Then, for any object i ∈ I with x ∈ Xyi , the distance of x to the hyperplane ω>x + β = 0 is given by y ( ω>xi + β ) ‖ω‖◦ , [192], where ‖ · ‖◦ denotes the dual of ‖ · ‖, i.”