On a conjecture about relative lengths

Abstract

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).

The relative lengths of sides of convex hexagons and octagons

“…The second part of this conjecture was proved in [2]. In this paper we prove the first part of this conjecture.…”

“…Take the lines through a, c, e parallel to the segments ce, ea, ac, respectively, the intersection points of these lines are denoted by a * , c * , e * (they are opposite to a, c, e, respectively, as shown in Figs. [1][2][3][4]. From the maximality of the area of T we conclude that the straight lines through the vertices parallel to the opposite sides of T are supporting lines of H. Therefore we know that b ∈ ae * c, d ∈ ca * e, and f ∈ ac * e. Case 1. f = f 0 .…”