等周不等式的探討

Abstract

主要是在探討R^2上的等周不等式，以及它在surface，R^n，manifold immersed in R^n上的變形！大綱如下： • 首先我用分析的手法論證R^2上的等周不等式 • 利用Gauss-Bonet定理探討等周不等式在surface in R^3的變形 • 利用Minkowski-Brunn Inequality 去論證R^n上的等周不等式 • 利用R^n上的等周不等式去討論R^n上的等周問題 • 利用適當的覆蓋定理去論證manifold immersed in R^n上的等周不等式 這篇論文主要是探討各個情況下等周不等式的樣子，所以對於regularity都假設得很好，以避開一些幾何測度論的問

Let C be a simple closed curve of length L on R^2, and Ω be the domain bounded by C of area A, then L^2 ≥ 4πA...... (∗) This is the simplest case of isoperimetric inequalities. We concentrate on the isoperimetric inequality (∗) and its extension. The contents in brief are as follows: • The classical case for curve in plane • Analogs of (∗) for domains on surfaces • Extensions of (∗) to domain in R^n • Solve the isoperimetric problem in R^n • Variants of (∗) for smooth manifolds immersed in R^n