Abstract
We provide, in a general setting without any algebraic and/or topological structure, a minimal axiomatic treatment of finite means which can be computed by a suitable weighted arithmetic mean. To this end, taking inspiration from Archimede’s theory of centre of gravity, we introduce, for non-null masses with finite support in an arbitrary non empty set Ω, the notion of quasi-barycentric mean, i.e. a map from the set of all such masses into Ω which satisfies the “intuitive” properties of consistency, shift-sensitivity and associativity (related to the barycentre of a system of material points). Moreover, we prove that it can be computed by the barycentre of finite distributions of masses in a (essentially unique) real vector space.
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