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Báo cáo toán học: "A Note about Bezdek’s Conjecture on Covering an Annulus by Strips"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Note about Bezdek’s Conjecture on Covering an Annulus by Strips. | A Note about Bezdek s Conjecture on Covering an Annulus by Strips Yuqin Zhang1 and Ren Ding2 1 Department of Mathematics Tianjin Univerity Tianjin 300072 China yqinzhang@163.com yuqinzhang@126.com 2 College of Mathematics Hebei Normal University Shijiazhuang 050016 China rending@hebtu.edu.cn rending@heinfo.net Submitted May 19 2007 Accepted Jun 3 2008 Published Jun 13 2008 Mathematics Subject Classifications 52C15 Abstract A closed plane region between two parallel lines is called a strip. Andras Bezdek posed the following conjecture For each convex region K there is an 0 such that if K lies in the interior of K and the annulus K K is covered by finitely many strips then the sum of the widths of the strips must be at least the minimal width of K. In this paper we consider problems which are related to the conjecture. 1 Introduction and Basic Definitions A closed plane region between two parallel lines at distance d is called a strip of width d. For each direction 0 0 0 a convex region M has two parallel supporting lines and the distance between them is denoted by 0 . The minimum 0 is called the minimal width of M. In the case of a triangle the minimal width is the altitude on the longest side. Let O denote the origin of the plane E2. For a given convex set K and 0 let K -- -- denote a homothetic copy of K consisting of all points X such that OX OY where Y 2 K. Tarski 5 conjectured and Bang 1 proved that if a convex region K can be covered by a finite collection of strips then the sum of the widths of the strips must be at least This research was supported by National Natural Science Foundation of China 10571042 10701033 10671014 . THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N19 1 the minimal width of K. Andras Bezdek 2 posed the following conjecture and proved two theorems Conjecture. 2 For each convex region K there is an 0 such that if K lies in the interior of K and the annulus K K is covered by finitely many strips then the sum of the widths of the strips .