{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22612"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22612","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Convex lattice polygons","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Alarcon, Eberth Guillermo, II"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Stolarsky, Kenneth B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:45:29Z","date_published":"2011-05-07T13:45:29Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Alarcon, Eberth Guillermo, II"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543512","(UMI)AAI9543512"],"render_values":[{"text":"AAI9543512","href":null,"code":true},{"text":"(UMI)AAI9543512","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22612","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Stolarsky, Kenneth B."]},{"key":"dc:creator","label":"Author","values":["Alarcon, Eberth Guillermo, II"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:45:29Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Alarcon, Eberth Guillermo, II"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543512","(UMI)AAI9543512","http://hdl.handle.net/2142/22612"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2011-05-07T13:45:29Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543512.pdf: 1613989 bytes, checksum: 4b28c01a20021092878677a14f47d561 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:58:48Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:41-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Convex lattice polygons"]}]}],"canonical_facts":{"dc:contributor":["Stolarsky, Kenneth B."],"dc:creator":["Alarcon, Eberth Guillermo, II"],"dc:date":["2011-05-07T13:45:29Z","10000-01-01","1995"],"dc:description":["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.","Made available in DSpace on 2011-05-07T13:45:29Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543512.pdf: 1613989 bytes, checksum: 4b28c01a20021092878677a14f47d561 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:58:48Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:41-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9543512","(UMI)AAI9543512","http://hdl.handle.net/2142/22612"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Alarcon, Eberth Guillermo, II"],"dc:subject":["Mathematics"],"dc:title":["Convex lattice polygons"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}