Local convexity properties of quasihyperbolic balls in punctured space (Q2481836)

In this paper, the author studies the metric space equipped with the quasihyperbolic distance and the results mainly deal with local convexity properties of the quasihyperbolic metric in the punctured space. Convexity and starlikeness of the quasihyperbolic balls are considered. Motivated by a question of \textit{M. Vuorinen} [Ponnusamy, S. (ed.) et al., Proceedings of the international workshop on quasiconformal mappings and their applications, December 27, 2005 -- January 1, 2006. New Delhi: Narosa Publishing House. 291--325 (2007, Zbl 1166.30013)] about the convexity of balls of small radii in metric spaces, the author provides an answer in a particular case.