A Chebyshev set and its distance function

[摘要] 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 Gateaux derivatives on X\C which are determined by the subdifferential partial derivativeparallel tox - (x) over bar parallel to for each x is an element of X\C and (x) over bar is an element of P-C(x) := {c is an element of C: parallel tox - cparallel to = dc(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 Frechet differentiable then C is convex iff d(C) is Frechet differentiable on X\C. If also X has a uniformly Gateaux differentiable norm then C is convex iff the Gateaux (Frechet) subdifferential partial derivative(-) d(C) (x) (partial derivative(F)d(C)(x)) is nonempty on X\C. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-12-01