On the smoothness of quasihyperbolic balls

Riku Klen*, Antti Rasila, Jarno Talponen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

Abstract

We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is C-1-smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of Gehring and Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls. We introduce an interesting application of quasihyperbolic metrics to equivalent renormings of Banach spaces. Several examples and illustrations are provided.

Original languageEnglish
Pages (from-to)439-452
Number of pages14
JournalANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA
Volume42
Issue number1
DOIs
Publication statusPublished - 2017
MoE publication typeA1 Journal article-refereed

Keywords

  • Quasihyperbolic metric
  • geodesics
  • uniqueness
  • smoothness
  • convexity
  • renorming
  • BANACH-SPACES
  • CONVEXITY PROPERTIES
  • UNIFORM DOMAINS
  • GEODESICS

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