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 Title: Convex lattice polygons Author(s): Alarcon, Eberth Guillermo, II Doctoral Committee Chair(s): Stolarsky, Kenneth B. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: This thesis deals with three main extremal problems on convex lattice polygons in the plane. A convex lattice polygon is the intersection of a compact convex set with the integer lattice (the set of all points with integer coordinates). Let P represent a convex lattice polygon.A fundamental concept is that of lattice diameter. The lattice diameter of P is the most (lattice) points on a line through P. A line containing maximally many points from P is also referred to as a lattice diameter.The first question I deal with is: given a fixed integer n, what is the largest area which a convex lattice polygon with lattice diameter n may have? I find precise answers for $n\le5$, and the answer within 2 (regardless of n) for $n\ge6$.Secondly, I demonstrate that, if P has lattice diameter $n\ge3$, then we can assume that all lines through P which contain n points have slope either 0, $\infty$, or $\pm$1. This work has its motivation in Tarski's "Plank Problem", solved in 1951 by T. Bang.Lastly, I consider the notion of local lattice diameters. The local lattice diameter of P at a point p is the most points from P on a line through p. If P contains at least 2 points, then certainly all local lattice diameters lie between 2 and the lattice diameter of P. The interesting question here is: how short (relative to the lattice diameter of P) can local lattice diameters be? For a compact convex set C in the plane, it is easy to see that any "local diameter" must be at least half as long as the (Euclidean) diameter of C. I show that convex lattice polygons exhibit similar behavior only if they satisfy a strict condition on the number of points they contain. As a last thought, I present an analysis of the distribution of local lattice diameters in convex lattice polygons.These results are compared with the case of compact convex sets in the plane, which serve as a familiar starting ground. While there are obvious differences with my results on point sets, some beautiful similarities become apparent. Issue Date: 1995 Type: Text Language: English URI: http://hdl.handle.net/2142/22612 Rights Information: Copyright 1995 Alarcon, Eberth Guillermo, II Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9543512 OCLC Identifier: (UMI)AAI9543512
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