Some remarks about closed convex curves

Abstract

We introduce a function w k (θ) for closed convex plane curves, and then prove a geometric inequality involving w k (θ) and the area A enclosed by the curve. As a by-product, we give a new proof of the classical isoperimetric inequality. Finally, we give some properties of convex curves with w k (θ) being constant and pose an open problem motivated by the elegant Blaschke―Lebesgue theorem.