[Packing and Covering with Centrally Symmetric Convex Discs]
by
Acknowledgement
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Abstract
In 1950, C.A. Rogers introduced and studied the simultaneous packing and covering constants for a convex body and obtained the first general upper bound. Afterwards, they have attracted the interests of many authors such as L. Fejes Toth, S.S. Ryskov, G.L. Butler, K. Boroczky, H. Horvath, J. Linhart and M. Henk since, besides their own geometric significance, they are closely related to the packing densities and the covering densities of the convex body, especially to the Minkowski-Hlawka theorem. However, so far our knowledge about them is still very limited. In this thesis we will determine the optimal upper bound of the simultaneous packing and covering constants for two-dimensional centrally symmetric convex domains, and characterize the domains attaining the upper bound. This thesis deals with arrangements of replicas of a plane convex body K. We prove certain sharp inequalities linking the packing and the covering densities, d(K) and ?(K), associated to this body. In particular we find the exact analytic description of the set of pairs (d(K), ? (K)) when K belongs to the class of centrally symmetric octagons.
Table of Contents
CHAPTER 1: INTRODUCTION6
The Brunn-Minkowski Theoretical Framework6
Deviation in Geometry9
High-Dimensional Geometry12
Centrally symmetric convex body18
Definition, notation and preliminaries19
Lemma 1.121
Lemma 1.221
Theorem 1 .121
Lemma 1.321
Proof22
Proof23
Theorem 1.225
Proof26
Corollary 1.129
Proof29
Corollary 1.229
Proof29
CHAPTER 2 : THE PROBLEM AND SUMMARY OF RESULTS33
Theorem 2.133
Proof of theorem 2.237
Proof of Theorem 2.343
Proof of Theorem 2.450
Proofs of Theorems 2.5 and 2.654
Conclusions and Open Problems54
CHAPTER 3: BRUNN-MINKOWSKI INEQUALITY PROOF56
Theorem 3.1 (Brunn)57
Theorem 3.2 (Brunn-Minkowski inequality)59
Theorem 3.3 (Isoperimetric inequality)59
Proof59
CHAPTER 4: THE QUEST FOR THE UPPER BOUND63
Theorem 4. 1(Kuperberg)65
Theorem 4.2 (Kuperberg)65
Theorem 4.366
The main construction66
Preliminary lemmas67
Proof67
Lemma 4.568
Proof68
Lemma 4.669
Proof70
Lemma 4.771
Proof71
Proof of theorem 4.372
Lemma 4.873
Proof73
CHAPTER 5: THE UPPER BOUND FOR A(K): CUTTING CORNERS78
Theorem 5.179
Theorem 5.279
An application: Spanning trees across barriers79
Theorem5. 380
Theorem 5.480
Related previous work on cuttings81
Preliminaries for Constructing Optimal Cuttings for Disks83
Subdivisions for Mutually Avoiding Disks84
Lemma 184
Proof of Lemma 184
CHAPTER 6: THE PROOF OF THE UPPER BOUND87
Lower Bounds87
Upper Bounds93
CHAPTER 7: CONCLUSION AND PLANS FOR THE FUTURE96
REFERENCES99
Chapter 1: Introduction
In next chapters, we consider arrangements of convex bodies in the Euclidean plane. A convex body is a compact convex set with nonempty interior; its area will be denoted by A( K). An arrangement of congruent replicas (translates) of a convex body K is a family A of convex bodies, each of which is congruent to (is translate of) K. The arrangement is a packing if its members' interiors are mutually disjoint, and it is a covering if the union of its members is the whole plane. For any pair of independent vectors u and v in E, the lattice generated by u and v is the set of vectors ...