A Chebyshev Set and its Distance Function

✍ Scribed by Zili Wu

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 140 KB
• Volume: 119
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.2002.3728

✦ Synopsis

We prove that in a Banach space X with rotund dual X n a Chebyshev set C is convex iff the distance function d C is regular on X =C iff d C admits the strict and G# a ateaux derivatives on X =C which are determined by the subdifferential @jjx À %

x xjj for each x 2 X =C and %

x x 2 P C ðxÞ :¼ fc 2 C : jjx À cjj ¼ d C ðxÞg: 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 n are Fr! e echet differentiable then C is convex iff d C is Fr! e echet differentiable on X =C: If also X has a uniformly G# a ateaux differentiable norm then C is convex iff the G# a ateaux (Fr! e echet) subdifferential @ À d C ðxÞ (@ F d C ðxÞ) is nonempty on X =C: # 2002 Elsevier Science (USA)