等周型不等式及其稳定性

The Isoperimetric Type Inequality and Its Stability

摘要: 本文建立了一个新的几何量——沿外法线向量的曲率轨迹，探讨了其几何意义及其与曲率中心轨迹的关系。并利用新的几何量建立了一组参数不等式：$\begin{array}{l}\alpha {\displaystyle {\int }_{\gamma }{\kappa }^2}ds+\beta {\displaystyle {\int }_0^{2\pi }{\rho }^2}\left(\theta \right)d\theta +\lambda {\displaystyle {\int }_0^{2\pi }{\rho }_{\beta }^2}\left(\theta \right)d\theta +\delta L^2+\sigma A\\ +\omega \left|\tilde{A}\right|+\mu \left|\widehat{A}\right|+\nu {\displaystyle {\int }_0^{2\pi }{\rho }_{\widehat{\beta }}^2}\left(\theta \right)d\theta +\zeta {\left({\rho }_M-{\rho }_m\right)}^2+\xi {\widehat{L}}^2\ge 0\end{array}$同时，本文还通过所建立的等周不等式推导出了一些新的几何Bonnesen型不等式，并研究了这些不等式的稳定性。

Abstract: In this paper, we establish a new geometric quantity-locus of curvature along outer normal vector. Its geometric meaning and its relationship with the curvature center locus are discussed. And, we use the new geometric quantity to establish a family of parametric inequalities:$\begin{array}{l}\alpha {\displaystyle {\int }_{\gamma }{\kappa }^2}ds+\beta {\displaystyle {\int }_0^{2\pi }{\rho }^2}\left(\theta \right)d\theta +\lambda {\displaystyle {\int }_0^{2\pi }{\rho }_{\beta }^2}\left(\theta \right)d\theta +\delta L^2+\sigma A\\ +\omega \left|\tilde{A}\right|+\mu \left|\widehat{A}\right|+\nu {\displaystyle {\int }_0^{2\pi }{\rho }_{\widehat{\beta }}^2}\left(\theta \right)d\theta +\zeta {\left({\rho }_M-{\rho }_m\right)}^2+\xi {\widehat{L}}^2\ge 0\end{array}$And we also use our isoperimetric inequalities to derive some new geometric Bonnesen-type in equalities. Furthermore, we investigate the stability property of such inequalities.