Two Typical Problems In Combinatorial Geometry

The famous book Unsolved Problems in Geometry [12] describes many challenging unsolved problems which attract great attention of mathematicians in the field of combinatorial geometry. In this disser-tation we discuss two typical problems in the combinatorial geometry mentioned in [12].Helly's theorem plays a very important role in combinatorial geome-try. The theorem states that for a finite family A of at least d+1 convex sets in R^d, if every d+1 members of A have a point in common, then there is a point common to all the members of A. There are many gener-alizations of Helly theorem. One of them is about geometric transversal. A geometric transversal is defined to be an affine subspace (such as a point, a line, a plane, or a hyperplane) intersecting every member of a given family.In Part I we discuss three kinds of such problems. In Chapter 2 we discuss point transversal to a family of translates of a convex sets in the plane, where we prove a famous conjecture of Griinbaum's by a concrete and straightforward method for some special cases. In Chapter 3 we obtain a new upper bound for "K-L problem" concerning line transversal to unit disks in the plane, where we prove that for a finite disjoint family F of unit disks in the plane, there is a line intersecting all but at most 5 members of F suppose that even' three of the members admit a line transversal. Thus we improve the latest upper bound 12. In Chapter 4 we obtain the Helly number for hyperplane transversal to translates of a convex cube in R^d. where we prove that the Helly number for suchfamilies is 5 when d = 2, and is greater than or equal to d + 3 when d≥3.At the same time, combinatorial geometry is replete with unresolved questions having to do with how things fit together. Many of these questions are easily understood and appear innocent and beguilingly simple, but most will probably remain unresolved for many years. The general situation with which we are concerned is the following. A figure F and a target set T are given in the plane. We say that the figure F fits in the target T, or, equivalently, the target T covers the figure F, when there is a rigid motion μ (an isometry of the plane) so that (F) T, i.e., if T has a subset congruent to F. It might happen that a specific figure F and target T are given, and one wants to know if the figure fits in the target. We call a problem of this kind a fitting problem. Or a family of figures might be given, and one seeks a target set, perhaps of prescribed shape, that covers every figure in the family and is small in some spec-ified sense. We refer to this sort of question as a covering problem. In Part II we solve four open problems in this area. In Chapter 6 we obtain the necessary and sufficient conditions for triangles to fit in squares and for triangles to fit in rectangles. In Chapter 7 we observe the smallest rectangular cover and triangular cover of any prescribed shape for any triangles of diameter one.