Combinatorial properties on convex polygons by a point set

我们在科研巨子（C）（2）中研究了凸多边形在给定点集上的一些组合性质。1.将点集划分为凸多边形：在1996年被DAM接受的论文“On a partition into convex polygons”中，我们介绍了一些划分性质：不相交划分、空划分和一般划分。2001年我们与本研究计画的研究者细野康之教授合作，成功地改善了不相交分割问题的界。关于空分拆，我们参加了2002年在中国举行的国际数学家大会，并对空分拆的界进行了改进。2002年我们在另一个国际会议上发表了关于这个问题的报告，并被采纳。2.关于具有指定内点数的凸多边形的存在性：关于这个问题的论文于2001年被DM采纳，并被采纳。 ...更多信息 K.Hosono和匈牙利数学家G.Karolyi等人对这类凸多边形的存在性作了更多的研究。2001年，我们围绕它进行了讨论，这份手稿被《离散几何》（A.Bezdek ed.）接受。2003.3.关于平面点集的不交凸k-阶数：我们估计了平面点集的凸k-阶数。特别是对于凸四边形，我们进行了研究，并于2001年发表了论文。本文介绍了对给定点集的有效划分方法，从而增加了上述问题1.4中凸四边形的数目。关于点的凸分解：我们寻求将给定点划分为凸胞元，使得这些胞元的并集构成一个简单多边形，并且每个点都在边界上。5.关于平面点集上面积最大的三角形问题：2002年参加国际会议后，我和一些同事在我们的私人研讨会上研究了这个问题。它是一个顶点在平面点集中的空三角形的最大面积与点的凸船体面积之比的问题。我们在2003年的印度尼西亚-日本联合会议上就这个问题发表了演讲，并提交了这份手稿。少

We have studied some combinatorial properties on convex polygons by a given point set in the research project ; giant-in-aid for scientific research (C)(2). In particular, we studied the following five problems.1.Partitioning a point set into convex polygons : In our paper "On a partition into convex polygons" which has accepted to DAM in 1996, we introduced some partitioning properties ; disjoint partition, empty partition and general partition. In 2001. we have succeeded to improve the bound for disjoint partition problem with Professor K.Hosono who is one of the investigators in this research project. About empty partition, we also improved the bounds after we attended the International Congress of Mathematicians 2002 in China. We gave the talk concerning this topic on the other international conference in 2002, and this paper was accepted.2.On the existence of a convex polygon with a specified number of interior points : The paper about this topics was accepted to DM in 2001, and w … More e studied more the existence of such convex polygon with K.Hosono and G.Karolyi who is a Hungarian mathematician. In 2001, we talked around it, and this manuscript was accepted to Discrete Geometry (A.Bezdek ed.) in 2003.3.On the number of disjoint convex k-gons for a planar point set : We estimated the number of convex k-gons for a planar point set. For a convex quadrilateral, in particular, we studied, and the paper accepted in 2001. In this paper, we introduced useful partitioning method for a given point set, and then the number of convex quadrilaterals is increase in the above problem 1.4.On convex decompositions of points : We seek a partition of a given point into convex cells such that the union of the cells forms a simple polygon and every point is on the boundary. The paper concerning this topics was accepted the international journal in 2001.5.On a triangle with the maximum area in a planar point set : After I attended the international conference in 2002, I studied this problem with some colleagues in our private seminar. It is the problem about the ratio between the maximum area of an empty triangle with vertices in a planar point set and the area of the convex hull of the points. We gave the talk about this problem in the Indonesia -Japan joint conference in 2003, and submitted this manuscript. Less