Abstract
We consider the following three functional equations123where f:R3→R.Considering their geometric meaning, equations (1) and (2) are known as ‘Cube’ and ‘Octahedron’ functional equations, respectively. Under the assumption of continuity, Haruki [2] has proved that (1) and (2) are equivalent. Etigson [3], has proved the equivalence of (1) and (2) under no regularity assumption. We will give here another proof. Also, under the assumption of continuity, Haruki has solved the ‘Cube’ functional equation. He gave the solution as a certain polynomial of fifth degree in x, y, z individually whose terms are the partial derivatives of a given polynomial.
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