概率测度在凸几何分析中的应用
发布时间：2018-09-02 09:06
【摘要】：本博士论文的研究内容隶属于几何分析中的凸体理论(简称凸几何或凸几何分析),该理论的核心内容是Brunn-Minkowski理论(又称为混合体积理论).本文主要致力于研究概率测度在凸几何分析中的应用,这是该领域研究的热点问题之一,本文主要涉及关于概率测度的质心不等式,关于Gaussian测度的Shephard问题,关于凸体不等式的函数化以及关于单调凸序列的Erd(?)s-Szekeres定理等问题的研究.质心体是凸几何中一个非常重要的几何概念,在信息论,分析学等领域有广泛的应用,而关于质心体的质心不等式是应用最广泛的仿射等周不等式之一.在本文第二章,我们首先给出了关于概率测度的广义(Orlicz)质心体的概念,说明新定义的广义质心体是一个凸体,然后用强大数定理与极限逼近的方法建立了相应的的质心不等式.当取特殊的密度函数和Orlicz函数时,广义质心体就变为经典的质心体,L_p质心体,Orlicz质心体以及平均带体等.特别的,本文结果统一了质心体与平均带体的定义,它们都是广义质心体的特殊情形.在第三章中,我们给出了关于概率测度的广义(Orlicz)质心体的非对称版本,并建立了相应的非对称质心不等式,方法依然是依赖于Paouris和Pivovarov等人的概率以及极限逼近的方法.当取特殊的密度函数和Orlicz函数时,一方面可以将非对称的经典(L_p,Orlicz)质心不等式推广到紧集上,另一方面还可以得到一些特殊的非对称凸体.对凸体的截面和投影的几何性质的研究具有非常重要的意义,是凸几何领域研究的热点问题之一,而与之相关的就是著名的Busemann-Petty问题和Shephard问题.在第四章中,我们讨论了关于Gaussian测度的Shephard问题,给出了当n≥3时Gaussian型Shephard问题解的一个反例,从而说明Gaussian型Shephard问题在n≥3时不成立,这与经典的Shephard问题是一致的.在第五章中,我们研究了C~+(S~(n-1))上的函数的一些性质和不等式.首先我们定义函数f的体积和表面积即为与f相关的Aleksandrov体的体积与表面积,得到函数f的表面积公式.接着通过讨论函数f与其极对偶函数f~°的关系,建立关于函数的Blaschke-Santal(?)型不等式.在第六章中,我们研究了单调凸序列的Erd(?)s-Szekeres定理,得到满足在任意n个实数组成的序列中选出r个元素构成单调凸子列或选出s个元素构成单调凹子列的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，
本文编号：2218893
[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，
本文编号：2218893
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