This article gives two constructions of a weighted mean which has a large domain, is affinely equivariant, has a locally high breakdown point and is locally uniformly linearizable. One construction is based on $M$-functionals with smooth defining $\psi$- and $\chi$ -functions which are used to control the weighting. The second construction involves a locally uniformly linearizable reduction of the data to a finite set of points. This construction has the advantage of computational speed and opens up the possibility of allowing the weighting to take the shape of the original data set into account. Its disadvantage lies in its inability to deal with large atoms. The aim of the locally uniform linearizability is to provide a stable analysis based on uniform asymptotics or uniform bootstrapping. The stability of the first construction is exhibited using different stochastic models and different data sets. Its performance is compared with three other functionals which are not locally uniformly linearizable.