Kohli, Dhruv; Rabin, Jeffrey M. Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity. (English) Zbl 1435.52005 J. Geom. 110, No. 2, Paper No. 40, 13 p. (2019). The authors study the behavior of convexity under expansions and contractions in the hyperbolic plane and in the sphere. They work with the Poincaré disk model of the hyperbolic plane and use the compactification \(\mathbb{C}\cup \{\infty\}\) to work with the sphere. The notion of expansion and contraction with respect to a point \(z\) in the hyperbolic plane and in the sphere is defined by using the idea of translating \(z\) to the origin, doing at the origin the dilation and translating back the result. The dilation at the origin is defined in the paper. By means of computations, the authors show that the expansion (resp. contraction) preserves the convexity of a set in the Poincaré disk (resp. in the sphere) if the transformation is done from a point inside the set but that the convexity may not be preserved in the other cases. Reviewer: Judit Abardia-Evéquoz (Frankfurt a. M.) Cited in 1 ReviewCited in 1 Document MSC: 52A55 Spherical and hyperbolic convexity Keywords:hyperbolic convexity; spherical convexity; preserving convexity; Poincaré disk; stereographic projection; dilation; radial expansion and contraction PDFBibTeX XMLCite \textit{D. Kohli} and \textit{J. M. Rabin}, J. Geom. 110, No. 2, Paper No. 40, 13 p. (2019; Zbl 1435.52005) Full Text: DOI arXiv References: [1] Beardon, A.F.: The Geometry of Discrete Groups. Springer, New York (1982) [2] Beardon, AF; Minda, D.; Ponnusamy, S. (ed.); Sugawa, T. (ed.); Vuorinen, M. (ed.), The hyperbolic metric and geometric function theory, 9-56 (2007), New Delhi · Zbl 1208.30001 [3] Ma, W., Minda, D.: Geometric properties of hyperbolic geodesics. In : Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (2007) · Zbl 1201.30055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.