Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 439-452

ON THE SMOOTHNESS OF QUASIHYPERBOLIC BALLS

Riku Klén, Antti Rasila and Jarno Talponen

University of Turku, Department of Mathematics and Statistics
FI-20014 Turku, Finland; riku.klen 'at' utu.fi

Aalto University, Department of Mathematics and Systems Analysis
P.O. Box 11100, FI-00076 Aalto, Finland; antti.rasila 'at' iki.fi

University of Eastern Finland, Department of Physics and Mathematics
Box 111, FI-80101 Joensuu, Finland; talponen 'at' iki.fi

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 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.

2010 Mathematics Subject Classification: Primary 58B10, 30C65; Secondary 46T05, 46B03.

Key words: Quasihyperbolic metric, geodesics, uniqueness, smoothness, convexity, renorming.

Reference to this article: R. Klén, A. Rasila and J. Talponen: On the smoothness of quasihyperbolic balls. Ann. Acad. Sci. Fenn. Math. 42 (2017), 439-452.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4226