On Combinatorial Properties for a given point set in Euclidean space

在研究项目中，我们研究了给定点集的一些组合性质；科学研究补助金(C)。特别地，我们研究了点集划分为不相交凸多边形的问题。我们在1996年国际期刊《离散应用数学》上发表的论文《On a partition into凸多边形》引入了三个划分性质；不相交分区、空分区和一般分区。此外，我们还证明了空间上不相交划分问题的结果，并被国际期刊《计算几何：理论与应用》所接受。1997年8月，我们在加拿大女王大学第九届计算几何会议上作了关于这些划分问题的报告。一些研究者对这些问题很感兴趣，我们讨论了一些相关的问题。1998年，我们与K.Hosono教授成功地改进了不相交分割问题的界，他是这个研究项目的主要研究者之一，因此我们在1998年1月向国际期刊提交了该论文。在本文中，我们介绍了一种新的方法。利用这种方法，我们能够估计出一个给定点集的不相交凸四边形的最大数目。这是新的和有趣的。匈牙利数学家g。karolyi教授，研究了一个相关的问题即凸四边形的一般划分，但是没有人研究这个问题。我们于1998年7月将这一问题的论文提交给国际期刊，并于1999年8月在麦吉尔大学第十届加拿大计算几何会议上作报告。另一方面，我们也与麦吉尔大学的K.Hosono教授和D.Avis教授一起研究了包含一定数量点的凸多边形的存在性问题。1997年12月，我们在美国数学学会和墨西哥数学学会第三次联合会议上做了讲座，并将论文提交给了纪念Helge Tverberg的《离散数学》特刊。并于1998年12月在日本离散与计算几何会议上作了关于这一问题的特例的报告，目前正在准备论文。少

We have studied some combinatorial properties for a given point set in the research project ; grant-in-aid for scientific research (C). In particular, we studied the problem on partitioning point set into disjoint convex polygons. Our paper "On a partition into convex polygons" which accepted the international journal : Discrete Applied Mathematics in 1996, introduced three partitioning properties ; Disjoint partition, Empty partition and General partition. Moreover we proved the result on the disjoint partition problem in space, it was also accepted the international journal : Computational Geometry : Theory and Applications. We gave the lecture concerning these partition problems on Ninth Canadian Conference on Computational Geometry at Queen's University in August 1997. Some researchers are interested in these problems and we talked about some related problems. In 1998, we have succeeded to improve the bound for disjoint partition problem with Professor K.Hosono who is one of the in … More vestigators in this research project, so we submitted such paper to the international journal in January 1998. In this paper, we introduced new method. Using this method, we were able to estimate the maximum number of disjoint convex quadrilaterals for a given point set. It is new and interesting. Professor G.Karolyi who is a Hungarian mathematician, studied the related problem that is the general partition into convex quadrilaterals, but nobody studies this problem. We submitted the paper concerning this problem to the international journal in July 1998 and gave the lecture on Tenth Canadian Conference on Computational Geometry at McGill University in August 199g.On the other hand, we also studied the related problem that is the existence of a convex polygon containing a specified number of points with Professor K.Hosono and Professor D.Avis at McGill University. We gave the lecture on Third Joint Meeting of the American Mathematical Society and the Sociedad Matematica Mexicana in December 1997 and submitted the paper to the special issue of Discrete Mathematics in honor of Helge Tverberg. Moreover, we gave the lecture concerning the special case of this problem on Japan Conference on Discrete and Computational Geometry '98 in December 1998 and we are preparing the new paper about it now. Less