On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.