Masatsugu Urabe

On a partition into convex polygons

Abstract In this paper we study the problem of partitioning point sets in the plane so that each equivalence class is a convex polygon with some conditions on the intersection properties of such sets. Let P be a set of n points in the plane. We define f(P) to be the minimum number of sets in a partition into disjoint convex polygons of P and F(n) as the maximum off(P), over all sets P of n points. We establish lower and upper bounds for F(n). We also estimate the maximum of the minimum number of sets in a partition into empty convex polygons, over all sets of n points. Finally, we consider the maximum of the minimum number of convex polygons which cover the n points set P, over all sets P of n points.

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 a counterexample to a conjecture of Mirzaian

Abstract Let L = { l 1 , l 2 ,…, l n } be a set of n pairwise disjoint line segments on the plane. A simple polygon Q is a circumscribing polygon of L if the vertices of Q are endpoints of segments in L and every segment in L is either an edge or an internal diagonal of Q . A Mirzaian conjectured that any L has a circumscribing polygon. In this note we present a counterexample to this conjecture.

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.

On circles containing the maximum number of points

Abstract We define B(x, y) to be the disk in the plane which has the points x, y as its diametral end points. Let ΠB(n) [or Π B (n)] be the largest number such that for every set [or every convex set] P of n points in R 2 , there exist two points x, y ϵ P for which B(x, y) contains Π B (n) [ or Π B (n)] points of P. We show that Π B (n) = Π B (n) = ⌈n/3⌉ + 1 .