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- Title
THE STRUCTURE OF CONTINUOUS RIGID FUNCTIONS OF TWO VARIABLES.
- Authors
Balka, Richárd; Elekes, Márton
- Abstract
A function ƒ : ℝn → ℝ is called vertically rigid if graph(cf) is isometric to graph(ƒ) for all c ≠ 0. In [1] we settled Jankovic's conjecture by showing that a continuous function ƒ : ℝ → ℝ is vertically rigid if and only if it is of the form a + bx or a + bekx (a; b; k ∈ ℝ). Now we prove that a continuous function ƒ : ℝ² → ℝ is vertically rigid if and only if, after a suitable rotation around the z-axis, ƒ (x; y) is of the form a+bx+dy, a+s(y)ekx or a+bekx +dy (a; b; d; k ∈ ℝ, k ≠ 0, s : ℝ → ℝ continuous). The problem remains open in higher dimensions.
- Subjects
GEOMETRIC rigidity; CONTINUOUS functions; MATHEMATICAL variables; FUNCTIONAL equations; MATHEMATICAL transformations; EXPONENTIAL functions
- Publication
Real Analysis Exchange, 2010, Vol 35, Issue 1, p139
- ISSN
0147-1937
- Publication type
Article
- DOI
10.14321/realanalexch.35.1.0139