We compute the robustness of Fermat-Weber points with respect to any finite gauge. We show a breakdown point of $1/(1+\sigma)$ where $\sigma$ 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.

翻译：本文计算了费马-韦伯点对于任意有限规范的鲁棒性。我们证明了其崩溃点为$1/(1+\sigma)$，其中$\sigma$是规范的非对称性度量。我们获得了定量结果，表明受污染的费马-韦伯点可能偏离真实值的程度，该偏离可用原始样本和污染部分的规模来表征。若该偏离仅取决于原始样本，则我们称该规范为“一致鲁棒的”。我们证明多面体规范是一致鲁棒的，而局部严格凸范数则不然；同时在二维空间中，任何一致鲁棒的规范必然是多面体的。