嵌入型星形平面曲线的反向等周型不等式

Reverse Isoperimetric-Type Inequality for Embedded Starshaped Plane Curves

摘要: 本文研究了嵌入星形平面曲线的等周型不等式，建立了一组涉及曲线长度、曲线所围区域面积、曲率中心轨迹面积，以及曲线曲率半径与曲率中心轨迹的积分的参数不等式，并通过简单的傅里叶级数证明。利用这些等周型不等式，本文还推导出了一些新的Bonnesen型不等式。

Abstract: In this paper, we deal with the isoperimetric-type inequalities for the embedded starshaped plane curves. In fact, we establish a family of parametric inequalities involving the following geometric functionals associated with the given starshaped curves with a simple Fourier series proof: Length of the curve, areas of the region included by the curve and the locus of curvature centers, and integral of the curvature radius of the curve and the locus of curvature centers. Using our isoperimetric-type inequalities, we also derive some new Bonnesen-type inequalities.