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https://hdl.handle.net/2142/22612
Description
Title
Convex lattice polygons
Author(s)
Alarcon, Eberth Guillermo, II
Issue Date
1995
Doctoral Committee Chair(s)
Stolarsky, Kenneth B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
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