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.