Convex and Discrete Geometry

Geir Agnarsson, Jill Bigley Dunham.* George Mason University, Fairfax, VA. Extremal coin graphs in the Euclidean plane. A coin graph is a simple geometric intersection graph where the vertices are represented by non-overlapping closed disks in the Euclidean plane and where two vertices are connected if their corresponding disks touch. The problem of determining the maximum number of edges of a unit coin graph on n vertices, where all the radii are of unit length, is well known and has a beautiful solution. In this talk we consider related extremal problems of coin graphs that satisfy certain natural conditions relating to the ratios of the possible radii of the coins of the graph. Further, we will explore the algebraic equations describing wheel graphs, as they relate to the maximum number of edges in our mentioned coin graphs. Javier Alonso, Horst Martini, Zokhrab Mustafaev.* University of Houston-Clear Lake, Houston, TX. On orthogonal chords in Minkowski spaces. It is known that a convex plate of diameter 1 in the Euclidean plane is of constant width 1 if and only if any two perpendicular intersecting chords have total length at least 1. We show that, in general, this result cannot be extended to normed (or Minkowski) planes when the type of orthogonality is defined in the sense of Birkhoff. Inspired by this, we present also further results on intersecting chords in normed planes that are orthogonal in the sense of Birkhoff and in the sense of James. András Bezdek*, Jan P Boronski, Wesley Brown, Braxton Carrigan, Matt Noble. Auburn University, Auburn, AL. On a new proof of the Malfatti’s problem. The following problem was posed by Malfatti in 1803: How to arrange in a given triangle three non-overlapping circles of greatest total area? Malfatti assumed that the solution would be obtained by three mutually touching circles each touching also two edges of the triangle (commonly called as Malfatti’s circles). Curiously, Malfatti had been wrong in his initial assumption. In 1930 Lob and Richmond observed that in an equilateral triangle the packing with one large inscribed triangle and two other inscribed in the remaining space is in fact better. In 1967 Goldberg outlined an argument, with graphical support, that Malfatti’s arrangement never solves the area maximizing problem. It was no sooner than in 1992, when Zalgaller and Los showed that greedy arrangement is always the best (i.e. where one chooses the circles in three steps, each time choosing a maximal possible one). In the present talk, by a simple non-analytic argument, we show that the solution to the original problem must be either the Malfatti arrangement or the greedy arrangement.