Inequalities for curvature integrals in Euclidean plane

Abstract

Abstract Let γ be a closed strictly convex curve in the Euclidean plane R2 $\mathbb{R}^{2}$ with length L and enclosing an area A, and A˜1 $\tilde{A}_{1}$ denote the oriented area of the domain enclosed by the locus of curvature centers of γ. Pan and Xu conjectured that there exists a best constant C such that L2−4πA≤C|A˜1|, $$\begin{aligned} L^{2}-4\pi A\leq C \vert \tilde{A}_{1} \vert , \end{aligned}$$ with equality if and only if γ is a circle. In this paper, we give an affirmative answer to this question. Moreover, instead of working with the domain enclosed by the locus of curvature centers we consider the domain enclosed by the locus of width centers of γ, and we obtain some new reverse isoperimetric inequalities.

Keywords

• Curvature
• Reverse isoperimetric inequality
• Oriented area
• Width function