Ball Versus Distance Convexity of Metric Spaces

We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when considering the Euclidean product. Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21794 Originally published at: Foertsch, T (2004). Ball versus distance convexity of metric spaces. Beiträge zur Algebra und Geometrie, 45(2):481-500. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 45 (2004), No. 2, 481-500. Ball Versus Distance Convexity of Metric Spaces Thomas Foertsch Institute for Mathematics, University of Zurich Winterthurerstrasse 190, 8057 Zurich, Switzerland e-mail: [email protected] Abstract. We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when considering the Euclidean product. MSC 2000: 53C70 (primary), 51F99 (secondary) We consider two different notions of convexity of metric spaces, namely (strict/uniform) ball convexity and (strict/uniform) distance convexity. Our main theorem states that (strict/uniform) distance convexity is preserved under a fairly general product construction, whereas we provide an example which shows that the same does not hold for (strict/uniform) ball convexity, not even when considering the Euclidean product. MSC 2000: 53C70 (primary), 51F99 (secondary)