The octahedron equation implies the cube equation: an elementary proof
type: Item
instance of: scholarly article
title: The octahedron equation implies the cube equation: an elementary proof
zbMATH Open document ID: 0608.39006
DOI: 10.1007/BF02188192
author: B. George
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137162
review text: It has been proved in the paper by \textit{J. Aczél}, the reviewer, \textit{M. A. McKiernan} and \textit{G. N. Sakovič} [ibid. 1, 37-53 (1968; Zbl 0157.461)] that the following two functional equations are equivalent without any regularity assumption: \[ (1)\quad f(x+t,y+t)+f(x-t,y+t)+f(x- t,y-t)+f(x+t,y-t)=4f(x,y) \] and (2) \(f(x+t,y)+f(x,y+t)+f(x-t,y)+f(x,y- t)=4f(x,y)\), where \(f:R^ 2\to R\) and x,y,t are real variables. It has been conjectured by D. Z. Djoković and the reviewer that the n- dimensional cube functional equation denoted by \((C_ n)\) which is a generalization of(1) and the n-dimensional octahedron functional equation denoted by \((O_ n)\) which is a generalization of (2) are equivalent for all \(n\geq 1\). \textit{L. Sweet} [ibid. 22, 29-38 (1981; Zbl 0475.39016)] proved that \((O_ n)\) implies \((C_ n)\) for all \(n\geq 1\). In the present note the authors give another proof of this theorem by using Newton's formulas on sums of powers of roots for algebraic equations.
Mathematics Subject Classification ID: 39B99
zbMATH DE Number: 3983780
zbMATH Keywords: n-dimensional cube functional equation
zbMATH Keywords: n-dimensional octahedron functional equation
zbMATH Keywords: Newton's formulas
reviewed by: Hiroshi Haruki
MaRDI profile type: Publication
cites work: On the generalized cube and octahedron functional equations
OpenAlex ID: W2012153023
Recommended article: Q4432122
Recommended article: Q3814149
Recommended article: The eight tetrahedron equations
Recommended article: The equivalence of two problems on the cube
Recommended article: Varignon's Theorem for Octahedra and Cross-Polytopes
Recommended article: Reduction of quad-equations consistent around a cuboctahedron I: Additive case
Recommended article: An excluded minor theorem for the octahedron
Recommended article: Invariant cubature formulas for the hyperoctahedron
Recommended article: The theorem of Minkowski for cubes
description: scientific article; zbMATH DE number 3983780