【摘要】：本博士論文的研究內容隸屬于幾何分析中的凸體理論(簡稱凸幾何或凸幾何分析),該理論的核心內容是Brunn-Minkowski理論(又稱為混合體積理論).本文主要致力于研究概率測度在凸幾何分析中的應用,這是該領域研究的熱點問題之一,本文主要涉及關于概率測度的質心不等式,關于Gaussian測度的Shephard問題,關于凸體不等式的函數(shù)化以及關于單調凸序列的Erd(?)s-Szekeres定理等問題的研究.質心體是凸幾何中一個非常重要的幾何概念,在信息論,分析學等領域有廣泛的應用,而關于質心體的質心不等式是應用最廣泛的仿射等周不等式之一.在本文第二章,我們首先給出了關于概率測度的廣義(Orlicz)質心體的概念,說明新定義的廣義質心體是一個凸體,然后用強大數(shù)定理與極限逼近的方法建立了相應的的質心不等式.當取特殊的密度函數(shù)和Orlicz函數(shù)時,廣義質心體就變?yōu)榻浀涞馁|心體,L_p質心體,Orlicz質心體以及平均帶體等.特別的,本文結果統(tǒng)一了質心體與平均帶體的定義,它們都是廣義質心體的特殊情形.在第三章中,我們給出了關于概率測度的廣義(Orlicz)質心體的非對稱版本,并建立了相應的非對稱質心不等式,方法依然是依賴于Paouris和Pivovarov等人的概率以及極限逼近的方法.當取特殊的密度函數(shù)和Orlicz函數(shù)時,一方面可以將非對稱的經典(L_p,Orlicz)質心不等式推廣到緊集上,另一方面還可以得到一些特殊的非對稱凸體.對凸體的截面和投影的幾何性質的研究具有非常重要的意義,是凸幾何領域研究的熱點問題之一,而與之相關的就是著名的Busemann-Petty問題和Shephard問題.在第四章中,我們討論了關于Gaussian測度的Shephard問題,給出了當n≥3時Gaussian型Shephard問題解的一個反例,從而說明Gaussian型Shephard問題在n≥3時不成立,這與經典的Shephard問題是一致的.在第五章中,我們研究了C~+(S~(n-1))上的函數(shù)的一些性質和不等式.首先我們定義函數(shù)f的體積和表面積即為與f相關的Aleksandrov體的體積與表面積,得到函數(shù)f的表面積公式.接著通過討論函數(shù)f與其極對偶函數(shù)f~°的關系,建立關于函數(shù)的Blaschke-Santal(?)型不等式.在第六章中,我們研究了單調凸序列的Erd(?)s-Szekeres定理,得到滿足在任意n個實數(shù)組成的序列中選出r個元素構成單調凸子列或選出s個元素構成單調凹子列的n=n(r,s)的最小值.
[Abstract]:The research content of this doctoral thesis belongs to convex body theory (convex geometry or convex geometry analysis) in geometric analysis. The core of this theory is Brunn-Minkowski theory (also called mixed volume theory). This paper focuses on the application of probabilistic measures in convex geometric analysis, which is one of the hot issues in this field. This paper mainly deals with the centroid inequality of probabilistic measures and the Shephard problem of Gaussian measures. In this paper, we study the functionalization of convex inequality and the Erd (?) s-Szekeres theorem for monotone convex sequences. Centroid is a very important geometric concept in convex geometry, which is widely used in the fields of information theory, analysis and so on. The centroid inequality of centroid is one of the most widely used affine isoperimetric inequalities. In the second chapter of this paper, we first give the concept of generalized (Orlicz) centroid about probability measure, and show that the new definition of generalized centroid is a convex body, and then establish the corresponding centroid inequality by using the strong number theorem and the method of limit approximation. When the special density function and Orlicz function are taken, the generalized centroid body becomes the classical centroid body, the Orlicz centroid body and the average banded body, etc. In particular, the definitions of centroids and average banded bodies are unified in this paper, which are special cases of generalized centroids. In chapter 3, we give the asymmetric version of the generalized (Orlicz) centroid of probability measure, and establish the corresponding asymmetric centroid inequality. The method is still dependent on the probability and limit approximation of Paouris and Pivovarov et al. When the special density function and the Orlicz function are taken, on the one hand, we can generalize the asymmetric classical mass center inequality to the compact set, on the other hand, we can obtain some special asymmetric convex bodies. It is of great significance to study the geometric properties of the section and projection of convex bodies. It is one of the hot issues in the field of convex geometry, and the related problems are the famous Busemann-Petty problem and Shephard problem. In chapter 4, we discuss the Shephard problem about Gaussian measure, give a counter example of the solution of Shephard problem of Gaussian type when n 鈮，

本文編號：2218893