TY - JOUR

T1 - Equivalence among various derivatives and subdifferentials of the distance function

AU - Wu, Zili

AU - Ye, Jane J.

N1 - Funding Information:
* Corresponding author. E-mail addresses: ziliwu@email.com (Z. Wu), janeye@math.uvic.ca (J.J. Ye). 1 The research of J.J. Ye was supported by NSERC and University of Victoria Internal Research Grant.

PY - 2003/6/15

Y1 - 2003/6/15

N2 - For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x ∈ X\C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x ∈ X \ C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.

AB - For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x ∈ X\C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x ∈ X \ C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.

KW - Distance function

KW - Proximal smoothness

KW - Proximal, Fréchet, Dini, and modified Dini subdifferentials

KW - Strict Gâteaux, and Fréchet derivatives

KW - Uniformly Gâteaux differentiable norm

UR - http://www.scopus.com/inward/record.url?scp=0037778834&partnerID=8YFLogxK

U2 - 10.1016/S0022-247X(03)00213-0

DO - 10.1016/S0022-247X(03)00213-0

M3 - Article

AN - SCOPUS:0037778834

SN - 0022-247X

VL - 282

SP - 629

EP - 647

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -