Inequalities for curvature integrals in Euclidean plane
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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2019-06-01
|
| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2116-5 |
| _version_ | 1818517076817477632 |
|---|---|
| author | Zengle Zhang |
| author_facet | Zengle Zhang |
| author_sort | Zengle Zhang |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-11T00:51:22Z |
| format | Article |
| id | doaj.art-7ac1eb012d6d429484a97ebf8eb0818b |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-11T00:51:22Z |
| publishDate | 2019-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-7ac1eb012d6d429484a97ebf8eb0818b2022-12-22T01:26:37ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-06-012019111210.1186/s13660-019-2116-5Inequalities for curvature integrals in Euclidean planeZengle Zhang0School of Mathematics and Finance, Chongqing University of Arts and SciencesAbstract 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.http://link.springer.com/article/10.1186/s13660-019-2116-5CurvatureReverse isoperimetric inequalityOriented areaWidth function |
| spellingShingle | Zengle Zhang Inequalities for curvature integrals in Euclidean plane Journal of Inequalities and Applications Curvature Reverse isoperimetric inequality Oriented area Width function |
| title | Inequalities for curvature integrals in Euclidean plane |
| title_full | Inequalities for curvature integrals in Euclidean plane |
| title_fullStr | Inequalities for curvature integrals in Euclidean plane |
| title_full_unstemmed | Inequalities for curvature integrals in Euclidean plane |
| title_short | Inequalities for curvature integrals in Euclidean plane |
| title_sort | inequalities for curvature integrals in euclidean plane |
| topic | Curvature Reverse isoperimetric inequality Oriented area Width function |
| url | http://link.springer.com/article/10.1186/s13660-019-2116-5 |
| work_keys_str_mv | AT zenglezhang inequalitiesforcurvatureintegralsineuclideanplane |