Abstract
In this paper, we assume that E is a real normed space of dimension at least two. The aim of this paper is to show that, for any convex domain in E, each quasigeodesic in this domain is quasiconvex in the norm metric. This result gives an affirmative answer to an open problem raised recently by Väisälä.
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Ahlfors L.V.: Quasiconformal reflections. Acta Math. 109, 291–301 (1963)
Beardon, A.F.: The Apollonian metric of a domain in \({\mathbb R^{n}}\). In: Quasiconformal Mappings and Analysis, pp. 91–108. Springer, Berlin (1998)
Broch, O.J.: Geometry of John Disks, Ph. D. Thesis, NTNU (2004)
Gehring F.W.: Uniform domains and the ubiquitous quasidisk. Jahresber. Deutsch. Math. Verein 89, 88–103 (1987)
Gehring F.W., Hayman W.K.: An inequality in thetheory of conformal mapping. J. Math. Pure Appl. 41(9), 353–361 (1962)
Gehring F.W., Osgood B.G.: Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36, 50–74 (1979)
Gehring F.W., Palka B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)
Hästö P.: The Apollonian metric: uniformity and quasiconvexity. Ann. Acad. Sci. Fenn. Ser. Math. 28, 385–414 (2003)
Heinonen J., Rohde S.: The Gehring-Hayman inequality for quasihyperbolic geodesics. Math. Proc. Cambridge Philos. Soc. 114, 393–405 (1993)
Kim K., Langmeyer N.: Harmonic measure and hyperbolic distance in John disks. Math. Scand. 83, 283–299 (1998)
Martio O.: Definitions of uniform domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 197–205 (1980)
Martio O., Sarvas J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 383–401 (1978)
Näkki R., Väisälä J.: John disk. Expo. Math. 9, 3–43 (1991)
Väisälä J.: Lectures on n-Dimensional Quasiconformal Mappings. Springer, Berlin (1971)
Väisälä J.: Uniform domains. Tohoku Math. J. 40, 101–118 (1988)
Väisälä J.: Free quasiconformality in Banach spaces. II. Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 255–310 (1991)
Väisälä J.: Relatively and inner uniform domains. Conformal Geom. Dyn. 2, 56–88 (1998)
Väisälä J.: The free quasiworld, Quasiconformal and related maps in Banach spaces. Banach Center Publ. 48, 55–118 (1999)
Väisälä J.: Quasihyperbolic geodesics in convex domains. Results Math. 48, 184–195 (2005)
Vuorinen, M.: Conformal Geometry and Quasiregular Mappings (Monograph, 208 pp.). In: Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)
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This paper was initiated during visit of M. Huang and X. Wang to IIT Madras and was completed during X. Wang’s visit to the Institute of Mathematics of Academia Sinica. These authors express their thanks to these two institutes for hospitality. The research was partly supported by NCET (No. 04-0783), NSFs of China (No. 10771059) and Tianyuan Foundation. The result of S. Ponnusamy was supported by NBHM, India.
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Huang, M., Ponnusamy, S. & Wang, X. The Quasiconvexity of Quasigeodesics in Real Normed Vector Spaces. Results. Math. 57, 239–256 (2010). https://doi.org/10.1007/s00025-010-0025-5
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DOI: https://doi.org/10.1007/s00025-010-0025-5