Whitney arcs and 1-critical arcs

Marianna Csörnyei; Jan Kališ; Luděk Zajíček

Fundamenta Mathematicae (2008)

  • Volume: 199, Issue: 2, page 119-130
  • ISSN: 0016-2736

Abstract

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A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that l i m y → x , y ∈ γ | f ( y ) - f ( x ) | / | y - x | = 0 for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.

How to cite

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Marianna Csörnyei, Jan Kališ, and Luděk Zajíček. "Whitney arcs and 1-critical arcs." Fundamenta Mathematicae 199.2 (2008): 119-130. <http://eudml.org/doc/282801>.

@article{MariannaCsörnyei2008,
abstract = {A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $lim_\{y→x, y∈γ\} |f(y)-f(x)|/|y-x| = 0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.},
author = {Marianna Csörnyei, Jan Kališ, Luděk Zajíček},
journal = {Fundamenta Mathematicae},
keywords = {Whitney curve; quasiarc},
language = {eng},
number = {2},
pages = {119-130},
title = {Whitney arcs and 1-critical arcs},
url = {http://eudml.org/doc/282801},
volume = {199},
year = {2008},
}

TY - JOUR
AU - Marianna Csörnyei
AU - Jan Kališ
AU - Luděk Zajíček
TI - Whitney arcs and 1-critical arcs
JO - Fundamenta Mathematicae
PY - 2008
VL - 199
IS - 2
SP - 119
EP - 130
AB - A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $lim_{y→x, y∈γ} |f(y)-f(x)|/|y-x| = 0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.
LA - eng
KW - Whitney curve; quasiarc
UR - http://eudml.org/doc/282801
ER -

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