On a conjecture about relative lengths

Abstract

We need some definitions from [1]. Let C ⊂ R2 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 λC(ab). Denote by λn the relative length of a side of the regular n-gon. For every ab ⊂ C we have |ab| ≤ |a′b′|, where a′b′ is the affine diameter parallel to ab, hence 0 < λn = |ab| |a′b′|/2 ≤ 2. For every regular triangle (or square), since its side length equals its corresponding affine diameter, λ3 = λ4 = 2. Let C = abcde be a regular pentagon with side length 1, join the points c and e, then we know that ab is parallel to ce and λ5 = λC(ab) = |ab| |ce|/2 = 1/ cos(π5 ) = √ 5 − 1 (see Fig. 1). Let C = abcdef be a regular hexagon with side length 1, join the points c and f , then ab is parallel to c f and λ6 = |ab| |c f |/2 = 1 (see Fig. 2). A side ab of a convex n-gon P is called relatively short if λP (ab) ≤ λn , and it is called relatively long if λP (ab) ≥ λn .