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.

The dual generalized Chernoff inequality for star-shaped curves

“…but γ is not a circle. This shows that the condition ρ k (θ) = C in Corollary 3.3 is necessary, which is different from the equality case of the Chernoff-Ou-Pan inequality [1,8].…”

“…where the equality holds if and only if α is a circle. Recently, Ou and Pan in [8] introduced the higher-order width function w k (θ) and got the Chernoff-Ou-Pan inequality (see [3]) as follows:…”

A unified approach to obtain discrete and smooth isoperimetric inequalities

“…Indeed, the Chernoff inequality can be regarded as a Wirtinger inequality for T 2 . By obtaining a first order Wirtinger-type inequality for T k , Ou and Pan [16] obtained a generalized version of the Chernoff inequality:…”

“…The equality holds if and only if γ is a circle. Ou and Pan [16] proved the following generalized version of the Chernoff inequality:…”

“…This is Ou-Pan [16] generalization of the Chernoff's inequality, which in turn is reduced to Chernoff inequality when k = 2.…”