Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 163-173

ON QUASIHYPERBOLIC GEODESICS IN BANACH SPACES

Antti Rasila and Jarno Talponen

Hunan Normal University, Department of Mathematics
Changsha, Hunan 410081, P.R. China; and
Aalto University, Institute of Mathematics
P.O. Box 11100, FI-00076 Aalto, Finland; antti.rasila 'at' iki.fi

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

Abstract. We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex domain \Omega in a Banach space such that there is no geodesic between any given pair of points x,y \in \Omega. In addition, we prove that if X is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of X, is smooth.

2010 Mathematics Subject Classification: Primary 30C65, 46G10, 58C20.

Key words: Quasihyperbolic metric, quasihyperbolic geodesic, uniform convexity, Banach space, C¹ smoothness, renormings, reflexive, Radon-Nikodym property, convex domain.

Reference to this article: A. Rasila and J. Talponen: On quasihyperbolic geodesics in Banach spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 163-173.

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doi:10.5186/aasfm.2014.3924