Gauge distances and median hyperplanes

A median hyperplane in d-dimensional space minimizes the weighted sum of th e distances from a finite set of points to it. When the distances from thes e 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 me thods, 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.