Kiyoshi Hosono

On the number of disjoint convex quadrilaterals for a planar point set

Abstract For a given planar point set P, consider a partition of P into disjoint convex polygons. In this paper, we estimate the maximum number of convex quadrilaterals in all partitions.

On the existence of a point subset with a specified number of interior points

Abstract An interior point of a finite point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k⩾1, let g(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g(k) interior points has a subset of points containing exactly k interior points. We prove that g(1)=1, g(2)=4, g(3)⩾8 , and g(k)⩾k+2, k⩾4 . We also give some related results.

On the minimum size of a point set containing two non-intersecting empty convex polygons

Let n (k, l) be the smallest integer such that any set of n (k, l) points in the plane, no three collinear, contains both an empty convex k -gon and an empty convex l -gon, which do not intersect. We show that n (3,5) = 10, 12 ≤ n (4,5) ≤ 14, 16 ≤ n (5,5) ≤ 20.

On the Existence of a Point Subset with 4 or 5 Interior Points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k + 1 interior points. We proved that h(3) =3 in an earlier paper. In this paper we prove that h(4) = 7.

Some combinatorial problems

Abstract There are many interesting and sophisticated problems posed in the IMO, Putnam and domestic Olympiads. Some of these problems have deep mathematical background, nice generalizations, and lead to new areas of research in combinatorics. We investigate several topics in this category and mention some results and open problems.

A Minimal Planar Point Set with Specified Disjoint Empty Convex Subsets

For a planar point set P in general position, an empty convex k -gon or a k -hole of P is a convex k -gon H such that the vertices of H are elements of P and no element of P lies inside H . Let n (k 1 ,k 2 , *** ,k l ) be the smallest integer such that any set of n (k 1 , *** ,k l ) points contains a k i -hole for each i , 1 ≤ i ≤ l , where the holes are pairwise disjoint. We evaluate such values. In particular, we show that n (1,2,3,4,5) = 15.

CONSTRUCTIONS FROM EMPTY POLYGONS

Let P denote a finite set of points, in general position in the plane. In this note we study conditions which guarantee that P contains the vertex set of a convex polygon that has exactly k points of P in its interior.

Note: On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most 10n-187 elements. In this paper, we improve this upper bound to @?75(n-3)@?+1.

On the visibility graph of convex translates

Abstract We show that the visibility graph of a set of non-intersecting translates of the same compact convex object in R 2 always contains a Hamiltonian path. Furthermore, we show that every other edge in the Hamiltonian path can be used to obtain a perfect matching that is realized by a set of non-intersecting lines of sight.

On the existence of a convex point subset containing one triangle in the plane

Let g(k) be the smallest integer such that every planar point set in general position with at least g(k) interior points has a convex subset with precisely k interior points. In this paper, we show that g(3)=8 if the point sets have no empty convex hexagons.