等周不等式

Isoperimetric Inequalities

麥富鈞(Fu-Chun Mai)

指導教授 ： 張樹城

國立清華大學/理學院/數學系/碩士(2006年)

摘要

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。

關鍵字

等周不等式 ； 變分法

並列摘要

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 .

並列關鍵字

Isoperimetric Inequalities ； variation ； Coarea Formula

參考文獻

[C] Issac Chavel (2001) Isoperimetric inequalities, Cambridge University Press.

[Do Carmo] Do Carmo, Manfredo Differential Curves and Surfaces. Prentice Hall, New Jersey, 1976

[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,