Symmetrization and sharp Sobolev inequalities in metric spaces.
Whitney arcs and 1-critical arcs
A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that 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...
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