有關歐氏空間中凸體的Brunn-Minkowski不等式

Abstract

凸幾何是一個研究凸函數和凸集的領域。'一集合是個凸集'是個很好的特性，足夠強大使我們可以推導出豐富的結果，又不會太難達成，所以其結果也可以適用在廣闊的情形。在凸幾何中，有一個不等式叫Brunn-Minkowski不等式，它給出兩個凸體的體積和它們的Minkowski和的體積的關係。 原始的Brunn-Minkowski不等式對所有的凸體都適用，而Böröczky，Lutwak，Yang和Zhang給出了一個針對原點對稱凸體的更強猜想，並且證明了在二維歐式空間的情況。 在這篇論文中 我們首先介紹凸幾何中一些基本的概念，然後是Böröczky，Lutwak，Yang和Zhang的論文中的工作，其中他們證明了log-Brunn-Minkowski不等式。

Convex geometry is a branch of geometry that studies convex functions and convex sets. Because of the strong property, convexity, this research area has many successful theories and applications. In convex geometry, there is an inequality concerning the relationship between the volumes of two convex bodies, and the volume of their Minkowski sum. This important inequality is called Brunn-Minkowski inequality. The classical Brunn-Minkowski inequality is valid for every two convex bodies. For origin-symmetric convex bodies, stronger inequalities are studied and conjectured by Böröczky, Lutwak, Yang and Zhang. In their work, these inequalities were proved for origin-symmetric convex bodies in two dimensional Euclidean space. In this thesis we will introduce some basic notions in convex geometry and then the work of Böröczky, Lutwak, Yang and Zhang, in which they proved the log-Brunn-Minkowski inequality.