## 平面闭曲线的Bonnesen型不等式

1 重庆师范大学数学科学学院 重庆 401331

2 上海立信会计金融学院统计与数学学院 上海 201620

## The Bonnesen-type Inequalities for Plane Closed Curves

Bin Rui,1, Wang Xingxing,2, Zeng Chunna,1

1 School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331

2 School of Mathematics and Statistics, Shanghai Lixin University of Accounting and Finance, Shanghai 201620

 基金资助: 国家自然科学基金重大专项.  12141101重庆英才青年拔尖计划.  CQYC2021059145重庆市自然科学基金.  cstc2020jcyj-msxmX0609重庆市自然科学基金.  cstc2019jcyj-msxmX0390重庆市留学人员创新创业支持计划.  cx2019155重庆市教育委员会科学技术研究项目.  KJQN201900530重庆市教育委员会科学技术研究项目.  KJZD-K202200509重庆市研究生科研创新项目.  CYS22556重庆师范大学研究生科研创新项目.  YKC21036

 Fund supported: the Major Project of NSFC.  12141101the Young Top-Talent Program of Chongqing.  CQYC2021059145the NSF of Chongqing.  cstc2020jcyj-msxmX0609the NSF of Chongqing.  cstc2019jcyj-msxmX0390the Venture Innovation Support Program for Chongqing Overseas Returnees.  cx2019155the Technology Research Foundation of Chongqing Educational Committee.  KJQN201900530the Technology Research Foundation of Chongqing Educational Committee.  KJZD-K202200509the Graduate Scientific Research Innovation Project of Chongqing.  CYS22556the Graduate Scientific Research Innovation Project of Chongqing Normal University.  YKC21036

Abstract

The isoperimetric inequality is one of the most classical geometric inequalities in differential geometry. The stability of isoperimetric genus can be characterized by Bonnesentype inequality and Bottema-type inequality. In this paper, via the method of differential geometry, Wirtinger inequality, Sachs inequality and divergence theorem and so on, we investigate the Bonnesen-type inequalities and Bottema-type inequalities for plane closed curves, and obtain a series of new Bonnesen-type inequalities and Bottema-type inequalities for curvature integration.

Keywords： Wirtinger inequality ; Sachs inequality ; Bonnesen-type inequality ; Bottema-type inequality

Bin Rui, Wang Xingxing, Zeng Chunna. The Bonnesen-type Inequalities for Plane Closed Curves. Acta Mathematica Scientia[J], 2022, 42(6): 1601-1610 doi:

## 1 引言

1920年前后, Bonnesen给出了一系列具有下列性质的不等式

$$${{L}^{2}}-4\pi A\ge B,$$$

(1) $B\ge 0;$ (2) $B=0$当且仅当$C$为圆.

(Bonnesen不等式) 欧氏平面${{{{\Bbb R}} }^{2}}$中简单闭曲线$C$所围成区域的面积$A$, 周长$L$满足

$$${{L}^{2}}-4\pi A\ge {{\pi }^{2}}{{({{r}_{e}}-{{r}_{i}})}^{2}},$$$

## 2 主要引理

$$$0\le \frac{{{L}^{3}}}{4{{\pi }^{2}}}-{{\int_{C}{\left| X \right|}}^{2}}{\rm d}s\le \frac{{{L}^{4}}}{64{{\pi }^{2}}}\left( \int_{C}{{{\kappa }^{2}}}{\rm d}s-\frac{4{{\pi }^{2}}}{L} \right),$$$

## 3 平面闭曲线的Bonnesen型不等式

$$${{L}^{2}}-4\pi A\ge \frac{2{{\pi }^{2}}}{L}{{\int_{C}{\left| X-(\frac{L}{2\pi })N \right|}}^{2}}{\rm d}s,$$$

由(2.4) 式可知

$$$\frac{{{L}^{2}}}{2\pi }={{\int_{0}^{2\pi }{\left( \frac{L}{2\pi } \right)}}^{2}}{\rm d}t=\int_{0}^{2\pi }{\left( {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}+{{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}} \right)}{\rm d}t.$$$

$$$2A={\int\!\!\!\int}_{\Omega }{{\rm div} X{\rm d}A=\int_{C}{\langle X, N \rangle{\rm d}s=\int_{C}{\langle X, -JT \rangle }}}{\rm d}s =\int_{0}^{2\pi }{(x\frac{{\rm d}y}{{\rm d}t}}-y\frac{{\rm d}x}{{\rm d}t}){\rm d}t.$$$

$\begin{eqnarray} \frac{{{L}^{2}}-4\pi A}{\pi } &=&2\left[ \int_{0}^{2\pi }{\left( {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}+{{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}} \right)}{\rm d}t-\int_{0}^{2\pi }{\left( x\frac{{\rm d}y}{{\rm d}t}-y\frac{{\rm d}x}{{\rm d}t} \right)}{\rm d}t \right] {}\\ &=&\int_{0}^{2\pi }{\left[ {{\left( x-\frac{{\rm d}y}{{\rm d}t} \right)}^{2}}+{{\left( y+\frac{{\rm d}x}{{\rm d}t} \right)}^{2}}+\left( {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}-{{x}^{2}} \right)+\left( {{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}}-{{y}^{2}} \right) \right]}{\rm d}t. {\qquad} \end{eqnarray}$

$$${{L}^{2}}-4\pi A=\pi \int_{0}^{2\pi }{\left[ {{\left( x-\frac{{\rm d}y}{{\rm d}t} \right)}^{2}}+{{\left( y+\frac{{\rm d}x}{{\rm d}t} \right)}^{2}}+\left( {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}-{{x}^{2}} \right)+\left( {{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}}-{{y}^{2}} \right) \right]}{\rm d}t.$$$

$\begin{eqnarray} &&\int_{0}^{2\pi }{\left[ \left( {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}-{{x}^{2}} \right)+\left( {{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}}-{{y}^{2}} \right) \right]}{\rm d}t {}\\ &=&\int_{0}^{2\pi }{\left[ {{\left( \frac{{\rm d}x}{{\rm d}t} \right)}^{2}}+{{\left( \frac{{\rm d}y}{{\rm d}t} \right)}^{2}}-\left( {{x}^{2}}+{{y}^{2}} \right) \right]}{\rm d}t=\int_{0}^{2\pi }{\left( X'{{(t)}^{2}}-X{{(t)}^{2}} \right)}{\rm d}t. \end{eqnarray}$

$C$为圆周时, 因为$t=\frac{2\pi }{L}s$, 可得

$X(t)$是周期为$2\pi$的连续函数. 则由Wirtinger不等式知

$$$\int_{0}^{2\pi }{\left( {{f}^{'}}{{(t)}^{2}}-f{{(t)}^{2}} \right)}{\rm d}t\ge 0,$$$

$$${{L}^{2}}-4\pi A\ge \pi \int_{0}^{2\pi }{\left[ {{\left( x-\frac{{\rm d}y}{{\rm d}t} \right)}^{2}}+{{\left( y+\frac{{\rm d}x}{{\rm d}t} \right)}^{2}} \right]}{\rm d}t.$$$

$\begin{eqnarray} \int_{0}^{2\pi }{\left[ {{\left( x-\frac{{\rm d}y}{{\rm d}t} \right)}^{2}}+{{\left( y+\frac{{\rm d}x}{{\rm d}t} \right)}^{2}} \right]}{\rm d}t &=&\frac{2\pi }{L}{{\int_{0}^{L}{\left( x-\frac{L}{2\pi }\frac{{\rm d}y}{{\rm d}s} \right)}}^{2}} +{{\left( y+\frac{L}{2\pi }\frac{{\rm d}x}{{\rm d}s} \right)}^{2}}{\rm d}s {}\\ &=&\frac{2\pi }{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s. \end{eqnarray}$

$$${{L}^{2}}-4\pi A\ge \frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X \right|}}^{2}}{\rm d}s-4\pi A,$$$

由(3.3) 式可知

$$$2A=\int_{C}{\langle X, N \rangle }{\rm d}s.$$$

$$$L=\int_{C}{\kappa \langle X, N \rangle }{\rm d}s.$$$

$$${{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s={{\int_{C}{\left| X \right|}}^{2}}{\rm d}s-\frac{2AL}{\pi }+\frac{{{L}^{3}}}{{{(2\pi )}^{2}}}.$$$

$$${{L}^{2}}-8\pi A=\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X \right|}}^{2}}{\rm d}s.$$$

$$${{\int_{C}{\left| X \right|}}^{2}}{\rm d}s\le \frac{{{L}^{3}}}{4{{\pi }^{2}}}.$$$

$\begin{eqnarray} {{L}^{2}}-8\pi A & \ge &\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-\frac{4{{\pi }^{2}}}{L}\frac{{{L}^{3}}}{4{{\pi }^{2}}} {}\\ &=&\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-{{L}^{2}}. \end{eqnarray}$

$$${{L}^{2}}-4\pi A\ge \frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-{{L}^{2}}+4\pi A.$$$

$\begin{eqnarray} \frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s &=&\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X \right|}}^{2}}{\rm d}s-\frac{4{{\pi }^{2}}}{L}\frac{2AL}{\pi }+\frac{4\pi {}^{2}}{L}\frac{{{L}^{3}}}{{{(2\pi )}^{2}}} {}\\ &=&\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X \right|}}^{2}}{\rm d}s-8\pi A+{{L}^{2}}. \end{eqnarray}$

## 4 平面闭曲线的Bottema型不等式

考虑到

$$${{L}^{2}}-8\pi A=\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X \right|^{2}}}}{\rm d}s.$$$

$$${{\int_{C}{\left| X \right|}}^{2}}{\rm d}s\ge \frac{{{L}^{3}}}{4{{\pi }^{2}}}-\frac{{{L}^{4}}}{64{{\pi }^{2}}}\left( \int_{C}{{{\kappa }^{2}}}{\rm d}s-\frac{4{{\pi }^{2}}}{L} \right).$$$

$\begin{eqnarray} {{L}^{2}}-8\pi A &\le& \frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-\frac{4{{\pi }^{2}}}{L}\left[ \frac{{{L}^{3}}}{4{{\pi }^{2}}}-\frac{{{L}^{4}}}{64{{\pi }^{2}}}\left( \int_{C}{{{\kappa }^{2}}}{\rm d}s-\frac{4{{\pi }^{2}}}{L} \right) \right] {}\\ &=&\frac{4{{\pi }^{2}}}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s-{{L}^{2}}+\frac{{{L}^{3}}}{16}\int_{C}{{{\kappa }^{2}}}{\rm d}s-\frac{{{\pi }^{2}}{{L}^{2}}}{4}. \end{eqnarray}$

$$${{L}^{2}}-4\pi A\ge 2{{\pi }^{2}}{{R}^{2}}+4\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A.$$$

$$${{L}^{2}}-4\pi A-4\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A\ge 2{{\pi }^{2}}{{R}^{2}}.$$$

$$$\frac{1}{2{{\pi }^{2}}}\left( {{L}^{2}}-4\pi A-4\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A \right)\ge {{R}^{2}}.$$$

$$${{R}^{2}}=\frac{1}{L}{{\int_{C}{\left| X-\left( \frac{L}{2\pi } \right)N \right|}}^{2}}{\rm d}s.$$$

$\begin{eqnarray} {{L}^{2}}-8\pi A &\le& 4{{\pi }^{2}}\left[\frac{1}{2{{\pi }^{2}}}\left({{L}^{2}}-4\pi A-4\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A\right)\right]-{{L}^{2}}+\frac{{{L}^{3}}}{16}\int_{C}{{{\kappa }^{2}}}{\rm d}s-\frac{{{\pi }^{2}}}{4}{{L}^{2}} {}\\ &=&\frac{4-{{\pi }^{2}}}{4}{{L}^{2}}-8\pi A-8\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A+\frac{{{L}^{3}}}{16}\int_{C}{{{\kappa }^{2}}}{\rm d}s. \end{eqnarray}$

$$${{L}^{2}}-4\pi A\le \frac{4-{{\pi }^{2}}}{4}{{L}^{2}}-4\pi A-8\pi {\int\!\!\!\int}_{C}{X\cdot }H{\rm d}A+\frac{{{L}^{3}}}{16}\int_{C}{{{\kappa }^{2}}}{\rm d}s.$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Burago Y , Zalgaller V . Geometric Inequalities. Heidelberg: Springer-Verlag, 1988

Chakerian G D .

The isoperimetric theorem for curves on minimal surfaces

Proc Amer Math Soc, 1978, 69 (2): 312- 313

Dierkes U , Hildebrandt S , Tromba A J . Regularity of Minimal Surfaces. Heidelberg: Springer-Verlag, 2010

Escudero C , Reventó A , Solanes G .

Focal sets in two-dimensional space forms

Pacific J Math, 2007, 233 (2): 309- 320

Gage M E .

An isoperimetric inequality with applications to curve shortening

Duke Math, 1983, 50 (4): 1225- 1229

Green M , Osher S .

Steiner polynomials, wulff flows, and some new isoperimetric inequalities for conver plane curves

Asian J Math, 1999, 3 (3): 659- 676

Hurwitz A .

Sur quelques applications géométriques des séries de Fourier(French)

Ann Sci École Norm, 1902, 19 (3): 357- 408

Kwong K K , Lee H .

Higher order Wirtinger-type inequalities and sharp bounds for the isoperimetric deficit

Proceedings of the American Mathematical Society, 2021, 149 (11): 4825- 4840

Li P , Richard S , Yau S T .

On the isoperimetric inequality for minimal surfaces

Ann Scula Norm Sup Pisa Cl Sci, 1984, 11 (4): 237- 244

Ma L , Zeng C N .

Remark on isoperimetric inequality in the wulff flow case

Acta Math Sci, 2015, 35A (2): 306- 311

Osserman R .

The isoperimetric inequality

Bull Amer Math Soc, 1978, 84, 1182- 1238

Osserman R .

Bonnesen-style isoperimetric inequality

Amer Math Month, 1979, 86 (1): 1- 29

Ren D L . Introduction to Integral Geometry. Shanghai: Shanghai Science and Technology Press, 1988

Sachs H .

Ungleichungen für Umfang, Flächeninhalt und Trägheitsmoment konvexer Kurven

Acta Mathematica Hungarica, 1960, 11 (1/2): 103- 115

Santaló L A . Integral Geometry and Geometric Probability. London: Addison-Wesley, 1976

Xu W X , Chang M .

Inverse Bonnesen type inequality on the constant curvature plane

Acta Mathematica Sinica, 2020, 63 (4): 309- 318

Zeng C N , Ma L , Zhou J Z , et al.

The Bonnesen isoperimetric inequality in a surface of constant curvature

Sci China Math, 2012, 55, 1913- 1915

Zeng C N , Zhou J Z , Yue S S .

The symmetric mixed isoperimetric inequality of two planar convex domains

Acta Mathematica Sinica, 2012, 55, 1- 8

Zhang X M .

Bonnesen-style inequalities and pseudo-perimeters for polygons

J of Geometry, 1997, 60, 188- 201

Zhang X M .

A Refinement of the discrete Wirtinger inequality

Journal of Mathematical analysis and applications, 1996, 200, 687- 697

Zhou J Z , Ren D L .

Geometric inequalities from integral geometry point of view

Acta Math Sci, 2010, 30 (5): 1322- 1399

Zhu B C , Xu W X .

Homogeneous integral of inner parallel bodies

Scientia Sinica Mathematica, 2016, 46 (6): 807- 816

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