Two Results on Covering an Annulus by Strips

Yuqin Zhang and Ren Ding
Department of Applied Mathematics, Institute of Beijing Technology,
Beijing 100081,
P.R.China

Abstract    
A closed plane region between two parallel lines is called a strip. Let O denote the origin of the plane E². For a given convex set K and >0, any copy of the set is called a homothetic copy of K, denoted by K. If K lies in the interior of K, K \K is called an annulus. In [2] 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 width of the strips must be at least the minimal width of K

We prove that the conjecture is true for each triangle. Let P(u,v,) be a parallelogram with two adjacent sides equal to u, v (u v) and the smaller angle , we also show that the conjecture is true for P(u,v,) with .

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References
1. T.Bang, A solution of the plank problem, Proc. Amer. Math. Soc. 2 (1951) 990-993.
2. A.Bezdek, Covering an annulusby strips, Discrete Comput. Geom. 30 (2003) 177-180. 3. H.G. Eggleston, On triangles circumscribing plane convex sets, J. London Math.Soc. 28 (1953) 36-46.