假設C為平面上一條簡單封閉總長為L的區線以及G為曲線C所圍出來的區域,G的面積為A， 則我們恆有L² - 4 π A大於等於0。以此不等式為基礎,本篇文章的焦點集中在一系列的等周不等式，那些等周不等氏可以看作L² - 4 π A大於等於0在高維度空間或是更一般情形的推廣。

Let C be a simple closed curve of length L in R² and G be the domain bounded by C of the area A, we have
(*) L² - 4 π A >= 0.
The purpose of the paper focuses on inequalities which can be regarded as generalizations of (*) and inequalities which imply isoperimetric inequalities for n-dimensional manifolds in .

Contents
Abstract
Acknowledgement
Introduction 1
1. Isoperimetric Inequality For R² 3
2. Bonnesen Type Inequalities 6
3. Isoperimetric Inequality for Surfaces 9
4. Isoperimetric Inequality For Rⁿ 11
4.1. The Isoperimetric Inequality For Minkowski
Content. 11
4.2. Hausdorff Metric 11
4.3. Brumm-Minkowski Inequality 14
5. Isoperimetric Inequality Involving Mean Curvature 17
5.1 Mean Curvature 17
5.2 First Variation Of Area. Radical Variation 17
5.3 Covering Lemma 20
5.4 Isoperimetric Inequalities Involving Mean
Curvature 21
6. Applications 24
6.1 Sobolev Constant In Rⁿ 24
6.2 An Upper Bound For Sobolev's Constant On Closed
Surfaces 26
Reference 29

[Bu] Yu.D.Burago & V.A.Zalgaller (1980). Geometric inequalities. Berlin: Springer-Verlag, 1988
[C] Issac Chavel (2001) Isoperimetric inequalities, Cambridge University Press.
[Do Carmo] Do Carmo, Manfredo Differential Curves and Surfaces. Prentice Hall, New Jersey, 1976
[F] H.Federer (1969). Geometric Measure Theory. New York: Springer-Verlag
[Har] Hartman, P(1964) Geodesic Parallel Coordinate In The Large. Amer. J. Math 86, 705-727.
[O1] R.Osserman (1978) The isoperimetric inequalities. Bull.Amer.Math.Soc., vol 84, p.1182-1238.
[O2] R.Osserman (1979) Bonnesen-style isoperimetric inequalities. Amer.Math.Monthly, vol 1, p. 1-29,
[Peter Li] Peter Li Lecture Notes on Geometric Analysis. Department of Mathematics . University of California 1992; Revised - August 15, 1996