A Chebyshev Set and its Distance Function

Abstract

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 :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)

Convex analysis in normed spaces and metric projections onto convex bodies

“…Metric projections have been widely studied through the past decades and in a number of contexts (see, e.g., [1], [4], [10] and [34], further references will be given throughout the text). A very special case is when one considers the metric projections onto a given convex body (i.e., a compact, convex set with non-empty interior) K in R n endowed with the standard distance given for the Euclidean norm (which we denote by | • |).…”

Chebyshev Sets

“…Remark 3.30. Several authors have approached the question of the convexity of Chebshev sets by considering the differentiability of the associated distance function; see[37,41,89].…”