A metric characterization of cells

Abstract

We examine finite dimensional compact convex metric spaces each having the property that the union of two line segments in the space, having more than one point in common, is a line segment. The question has been asked (Borsuk; Bing) whether each such space is a cell. The answer is yes if the dimension of the space is ≤ 2 \leq 2 (Lelek and Nitka) or 3 (Rolfsen). Here we provide an affirmative answer for arbitrary finite dimension provided the space has the additional property that the join of any point to any line segment in the space is a convex set.