Combinatorial properties of weighted point sets in the plane

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540144
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tokai University
• Principal Investigator: Sakai Toshinori 東海大学, 高輪教養教育センター, 教授 (20267842)
• Research Collaborator: URRUTIA Jorge メキシコ国立自治大学, 数学研究所, 専任上級研究員レベルC ; HURTADO Ferran カタルーニャ工科大学(スペイン), 応用数学科II, 教授
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 重み付き点集合 / 単調多角形 / 単調無交差道 / 島

Outline of Final Research Achievements

Let P be a set of n points in general position in the plane, with weights 1, 2, ..., n. A monotonic path is a path connecting points of P along which the weights of its vertices monotonically increase or decrease. A monotonic polygon is a simple polygon whose perimeter consists of a non-crossing monotonic path and an edge. A subset S of P is called an island if S consists of the points of P contained in the convex hull of S. We studied the problems concerning the maximum number of vertices contained in monotonic polygons or non-crossing monotonic paths; the maximum number of the sums of weights of vertices of monotonic polygons; the number of faces contained in a decomposition of the convex hull of P into monotonic polygons; the existence of an island with weight close to a given number; the existence of disjoint islands each of which has weight close to a given number; and so on. We also obtained some results concerning the islands of weighted point sets in d-dimensional space.

Free Research Field

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