Abstract
Let γ be a closed strictly convex curve in the Euclidean plane mathbb{R}^{2} with length L and enclosing an area A, and 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 \t\t\tL2−4πA≤C|A˜1|,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} L^{2}-4\\pi A\\leq C \\vert \\tilde{A}_{1} \\vert , \\end{aligned}$$ \\end{document} 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.
Highlights
L2 – 4π A ≤ C|A 1|, with equality if and only if γ is a circle
Corollary 1 ([18]) Let γ be a C2 closed and strictly convex curve with length L in the Euclidean plane, κ be the curvature of γ, A the area enclosed by γ and A 1 the oriented area enclosed by β
Corollary 2 Let γ be a C2 closed and strictly convex curve with length L in the Euclidean plane, κ be the curvature of γ, A the area enclosed by γ and A 1 the oriented area enclosed by β
Summary
L2 – 4π A ≤ C|A 1|, with equality if and only if γ is a circle. In this paper, we give an affirmative answer to this question. Lemma 2 ([19]) Let γ be a C2 closed and strictly convex curve in the Euclidean plane R2 enclosing a domain D of area A.
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