A1 Refereed original research article in a scientific journal
ON THE SMOOTHNESS OF QUASIHYPERBOLIC BALLS
Authors: Riku Klén, Antti Rasila, Jarno Talponen
Publisher: SUOMALAINEN TIEDEAKATEMIA
Publication year: 2017
Journal: Annales Academiae Scientiarum Fennicae. Mathematica
Journal name in source: ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA
Journal acronym: ANN ACAD SCI FENN-M
Volume: 42
Issue: 1
First page : 439
Last page: 452
Number of pages: 14
ISSN: 1239-629X
DOI: https://doi.org/10.5186/aasfm.2017.4226
Web address : http://www.acadsci.fi/mathematica/Vol42/vol42pp439-452.pdf
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.
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.
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