On the smoothness of quasihyperbolic balls

Riku Klén, Antti Rasila, Jarno Talponen

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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 C1-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
Issue number1
StatePublished - 2017
Externally publishedYes


  • Convexity
  • Geodesics
  • Quasihyperbolic metric
  • Renorming
  • Smoothness
  • Uniqueness


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