Convexity Minimizes PseudoTriangulations Oswin Aichholzer #+ Franz Aurenhammer # Hannes Krasser ## Bettina Speckmann Summary: Convexity Minimizes Pseudo­Triangulations Oswin Aichholzer #+ Franz Aurenhammer # Hannes Krasser ## Bettina Speckmann § Abstract The number of minimum pseudo­triangulations is minimized for point sets in convex position. 1 Introduction A pseudo­triangle is a planar polygon with exactly three convex vertices, called corners. Three reflex chains of edges join the corners. Let S be a set of n points in general position in the plane. A pseudo­ triangulation for S is a partition of the convex hull of S into pseudo­triangles whose vertex set is S. A pseudo­triangulation is called minimum if it consists of exactly n - 2 pseudo­triangles (and 2n - 3 edges), the minimum possible. Each vertex of a minimum pseudo­ triangulation is pointed, that is, its incident edges span a convex angle. In fact, minimum pseudo­triangulations can be characterized as maximal planar straight­line graphs where each vertex is pointed [15]. Pseudo­triangulations, also called geodesic triangula­ tions, have received considerable attention in the last