Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity (Q2329375)

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.