On Combinatorial Properties for a given point set in Euclidean space

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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