Chisini mean

In mathematics, a function f of n variables

x1, ..., xn

leads to a Chisini mean M if for every vector <x1, ..., xn>, there exists a unique M such that[1]

f(M,M, ..., M) = f(x1,x2, ..., xn).

The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants.

While Oscar Chisini was arguably the first to deal with "substitution means" in some depth in 1929.[1], the idea of defining a mean as above is quite old. For instance, it can be found on pag. 35 of Vol. XV of the Penny Cyclopaedia (1839) in the lemma "Mean" by Augustus De Morgan .


See also

References
  1. Graziani, Rebecca; Veronese, Piero (2009). "How to Compute a Mean? The Chisini Approach and Its Applications". The American Statistician. 63 (1): 33–36. JSTOR 27644090.


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