## Abstract

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

Original language | English |
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Pages (from-to) | 181-192 |

Number of pages | 12 |

Journal | Journal of Approximation Theory |

Volume | 119 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Dec 2002 |

Externally published | Yes |