Doliwka and Lassak proved that if n ≤ 5 then every convex n-gon has a relatively short side and a relatively long side. In this paper we prove that this is true for n ≤ 5 only. More precisely, for any n ≥ 6, there exist n-gons without any relatively short sides.
Doliwka and Lassak proved that every convex pentagon must have both relatively short and long sides and showed that there exist convex hexagons without any relatively short sides. They conjectured that every convex hexagon has a relatively long side. We prove that this conjecture is true. We also show that every convex octagon has a side whose relative length is at most 1 and this bound is asymptotically tight.Keywords Affine diameter · Relative length · Relatively short side · Relatively long side AMS Subject Classification 52A10 · 52A40 · 52C15We need some definitions in [1]. Let C ⊂ R 2 be a convex body. A chord of C is called an affine diameter of C if there is no longer parallel chord in C. By |pq| we denote the length of the line-segment pq. For a chord ab of C, the ratio λ C (ab) = |ab|where a b is an affine diameter of C parallel to ab, is called the relative length of ab with respect to C.Denote by λ n the relative length of a side of the regular n-gon. Clearly, λ 6 = 1. A side ab of a convex n-gon P is called relatively short if λ P (ab) ≤ λ n and it is
Abstract. In this paper we prove two results about tilings of orthogonal polygons.(1) Let P be an orthogonal polygon with rational vertex coordinates and let R(u) be a rectangle with side lengths u and 1. An orthogonal polygon P can be tiled with similar copies of R(u) if and only if u is algebraic and the real part of each of its conjugates is positive; (2) Laczkovich proved that if a triangle ∆ tiles a rectangle then either ∆ is a right triangle or the angles of ∆ are rational multiples of π. We generalize the result of Laczkovich to orthogonal polygons.
in Shijiazhuang (China), where he is now associate professor of mathematics. His research interests focus primarily on discrete, convex, and combinatorial geometry. Ren Ding is professor of mathematics and supervisor of the Ph.D. programs at Hebei Normal University in Shijiazhuang (China). His research interests focus primarily on discrete, convex, and combinatorial geometry. We need some definitions from [1]. Let C ⊂ R 2 be a convex body. A chord pq of C is called an affine diameter of C, if there is no longer parallel chord in C. The ratio of |ab| to 1 2 |a b |, where a b is an affine diameter of C parallel to ab, is called the C-length of ab, or the relative length of ab, if there is no doubt about C. We denote it by λ C (ab).
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