A Characterization of Weighted Arithmetic Means

We prove, among other things, that the set of weighted arithmetic means is identical with the set of functions $f:R^n \to R$ satisfying (i) $\min \{ x_j \}\leqq f ( x_1 ,x_2 , \cdots ,x_n )\leqq \max \{ x_j \}$ and (ii) for $k = 2,3:\sum _{i = 1}^k x_{ij} = s( j = 1,2, \cdots ,n ) \Rightarrow \sum _{i = 1}^k f( x_{i1} ,x_{i2} , \cdots ,x_{in} ) = s$.