Abstract
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.
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References
Durier, R., and Michelot, C., Geometrical Properties of the Fermat-Weber Problem, European Journal of Operational Research, Vol. 20, pp. 332-343, 1985.
Michelot, C., The Mathematics of Continuous Location, Studies in Locational Analysis, Vol. 5, pp. 59-83, 1993.
Rockafellar, T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Thach, P. T., The Design Centering Problem as a DC Programming Problem, Mathematical Programming, Vol. 41, pp. 229-248, 1988.
Mangasarian, O. L., Arbitrary Norm Separating Plane, Operations Research Letters, Vol. 24, pp. 15-23, 1999.
Melachrinoudis, E., An Analytical Solution to the Minimum L p-Norm of a Hyperplane, Journal of Mathematical Analysis and Applications, Vol. 211, pp. 172-189, 1997.
Drezner, Z., and Wesolowsky, G. O., The Asymmetric Distance Location Problem, Transportation Science, Vol. 23, pp. 201-207, 1989.
SchÖbel, A., Locating Lines and Hyperplanes, Kluwer Academic Publishers, Dordrecht, Holland, 1998.
Norback, J. P., and Morris, J. G., Fitting Hyperplanes by Minimizing Orthogonal Deviations, Mathematical Programming, Vol. 19, pp. 102-105, 1980.
Avriel, M., Diewert, W. E., Schaible, S., and Zhang, I., Generalized Concavity, Plenum Press, New York, NY, 1988.
Carrizosa, E., and Plastria, F., Dominators for Multiple-Objective Quasiconvex Maximization Problems, Journal of Global Optimization, Vol. 18, pp. 35-58, 2000.
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Plastria, F., Carrizosa, E. Gauge Distances and Median Hyperplanes. Journal of Optimization Theory and Applications 110, 173–182 (2001). https://doi.org/10.1023/A:1017551731021
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DOI: https://doi.org/10.1023/A:1017551731021