On Relatively Short Sides of Convex Hexagons

Abstract

Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.