Breakdown points of Fermat–Weber problems under gauge distances

Abstract We compute the robustness of Fermat–Weber points with respect to any finite gauge. We show a breakdown point of $$1/(1+\sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> is the asymmetry measure of the gauge. We obtain quantitative results indicating how far a corrupted Fermat–Weber point can lie from the true value in terms of the original sample and the size of the corrupted part. If the distance from the true value depends only on the original sample, then we call the gauge ‘uniformly robust.’ We show that polyhedral gauges are uniformly robust, but locally strictly convex norms are not, while in dimension 2 any uniform robust gauge is polyhedral.