Equivalence among various derivatives and subdifferentials of the distance function

Zili Wu*, Jane J. Ye

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)629-647
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 15 Jun 2003
Externally publishedYes


  • Distance function
  • Proximal smoothness
  • Proximal, Fréchet, Dini, and modified Dini subdifferentials
  • Strict Gâteaux, and Fréchet derivatives
  • Uniformly Gâteaux differentiable norm

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