Proportion functions in three dimensions (Q1121467)

A functional equation approach to the construction and characterization of proportion functions on three-dimensional boxes is presented. Let \(D=(0,+\infty)\) and \(I=[1,+\infty)\); a proportion function in three dimensions is a function \(f:D^ 3\to I\) which is symmetric, normalized, i.e. \(f(x,x,x)=1\), and satisfies a condition of the form \(f(x,y,z)=f(\alpha (x,y,z))\) for all mappings \(\alpha:D^ 3\to D^ 3\) belonging to a fixed set B of bijections of \(D^ 3.\) In the first part of the paper the proportion functions are completely described when B is the set of all homotheties. In the second part, after giving two possible definitions of reciprocal box, the functions which are symmetric, normalized and constant on reciprocal boxes are characterized.