在第一節到第五節，我們介紹一些重要的曲線幾何量，如彎曲，扭轉，支撐函數和寬度。在這篇論文中接下來的部分，我們討論一些著名的幾何不等式，如等周不等式，Gage等周不等式和Wirtinger不等式。此外，我們引入了曲線的一個重要思想 - 平行曲線。使用平行曲線的想法，我們可以得到本文的我們最終的目標 - 熵估計。

In the first five sections we introduce some important geometric quantities of a curve, such as curvature, torsion, the support function and the width. In the rest part of this thesis we derive some famous geometric inequalities such as the isoperimetric inequality, Gage’s isoperimetric inequality and Wirtinger inequality. Also we introduce an important idea for a curve – the parallel curve. Using the idea of the parallel curve we can obtain our final goal of this thesis – the entropy estimate.

1. The Curve
2. The Application of Hölder and Andrews Inequality
3. The Support Function
4. Width Estimate of a Convex Closed Curve
5. Entropy Estimate of the Width
6. Gage’s Isoperimetric Inequality
7. The Result of Dohmen, Giga, and Mizoguchi
8. Parallel Curve of a Convex Closed Curve
9. Reference

1. Manfredo P. do Carmo,Differential geometry of curves and
surfaces ,1976,Prentice-Hall.

2. M. Gage, An isoperimetric inequality with applications to
curve shortening, Duke Math.J., 50 (1983), 1225-1229.

3. Rolf Schneider, Convex Bodies: The Brunn-Minkowski Theory,
2008,Combridge University