A Chebyshev set and its distance function

Zili Wu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


We prove that in a Banach space X with rotund dual X* a Chebyshev set C is convex iff the distance function dC is regular on X\C iff dC admits the strict and Gâteaux derivatives on X/C which are determined by the subdifferential ∂ x - x̄ for each x ∈ X\C and x̄ ∈ PC(x) := {c ∈ C: x - c = dC(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff PC is continuous. If the norms of X and X* are Fréchet differentiable then C is convex iff dC is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂-dC (x) (∂FdC(X)) is nonempty on X\C.

Original languageEnglish
Pages (from-to)181-192
Number of pages12
JournalJournal of Approximation Theory
Issue number2
Publication statusPublished - 1 Dec 2002
Externally publishedYes

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