摘要
本博士论文主要研究L_p-Brunn-Minkowski理论中的一些极值问题。本文首先介绍了所属学科的发展历程、研究现状和主要的代表人物以及作者的主要工作。接着研究了关于广义的投影体、相交体、质心体的单调性，然后重点研究了拟L_p-相交体，对偶L_p-John椭球和迷向L_p-表面积测度等。
作者取得的主要研究成果是：
(1) 关于投影体、相交体、质心体的单调性问题是凸体几何中最基本而又相当重要的问题，其中关于投影体和相交体的单调性问题分别是著名的Shephard问题和Busemann-Petty问题。我们将原有的结果推广到广义的投影体、相交体、质心体上，其中广义质心体是在本文中首次定义。
(2) 给出了拟L_p-相交体的定义并得到了拟L_p-Busemann相交不等式，得到了关于拟L_p-相交体的对偶Brunn-Minkowski不等式，考虑了它的单调性，推广到混合的拟L_p-相交体后得到了关于混合拟L_p-相交体的Aleksandrov-Fenchel不等式。并利用Aleksandrov-Fenchel不等式给出了一个唯一性定理。
(3) 对于p≥1，获得了一族对偶L_p-John椭球(?)_pK，这族椭球包括了两个在凸体几何与局部理论中都相当重要的椭球：L(o/¨)wner椭球(?)K和Legendre椭球Γ_2K，事实上，有(?)_∞K=(?)K,(?)_2K=Γ_2K。并且证明了对偶L_p-John椭球与L_p-质心体之间存在着John包含关系。这个结果与Lutwak，Yang和Zhang的《L_p-John椭球》形成了一种完美的对称。
(4) 应用L_p-John椭球及对偶L_p-John椭球的性质得到了一系列关于L_p-投影体、L_p-质心体的体积不等式，如L_p-Petty投影不等式和L_p-质心体不等式的逆向形式的不完全精确形式，并且得到了L_p-John椭球的另一种形式的包含关系，另外我们利用John基给出了L_p-型Loomis-Whitney不等式以及Pythagorean不等式。
(5) 研究了迷向L_p-表面积测度，证明了相同体积凸体的L_p-表面积在仿射变换下达到最小当且仅当此凸体的L_p-表面积测度是迷向的，对于L_p-表面积迷向的凸体将其L_p投影体的极体按其L_p-表面积给出了上下界估计，并且得到了L_p-等周不等式及其逆向形式。对于1≤p≤2时给出了L_p-表面积迷向位置的稳定性。
This Ph. D. dissertation sketches firstly the growing history, researching status, main represent figures in the researching branch and the author's research work; the following, it studies the monotone property of the generalized projection bodies , intersection bodies and centroid bodies; then, it studies emphasisly the quasi L_p-intersection bodies, the dual L_p-John ellipsoids, the isotropic L_p-surface area measure and so on.
The main results given by the author are as follows:
(i) The monotone property of the projection bodies intersection bodies and centroid bodies is the fundamental property in convex geometry. In fact, the monotone property of the projection bodies and the intersection bodies are the well known Shephard problem and the Busemann-Petty problem respectively. We established the monotone property for the generalized projection bodies intersecdtion bodies and centroid bodies. The generalized centroid bodies was first defined here.
(ii) We defined the quasi L_p-intersection bodies and established the L_p-Busemann intersection inequality. We also obtained the dual Brunn-Minkowski inequality for the quasi L_p-intersection bodies. After generalized the notion of quasi L_p-intersection bodies to that of mixed quasi L_p-intersection bodies, we given the Aleksandrov-Fenchel inequality and an unique theorm.
(iii) Given a convex body K, for p ≥ 1, we proved that there exists a family of ellipsoids E_pK such that the classical Lowner ellipsoid JK and the Legendre ellipsoid Γ_2K are the special cases of this family(p = ∞ and p = 2). This result is a perfect dual form of the 《L_p-John ellipsoids 》given by Lutwak, Yang and Zhang.
(iv) Using the properties of L_p-John ellipsoids and dual L_p-John ellipsoids, we obtained a lot of isopermetric inequalities for L_p-projection bodies , L_p-intersection bodies, for example, incompletely exact forms of L_p-Petty projection inequality and the inverse form of the L_p-centroid inequality. Moreover, we got an inclusion of the Lp-John ellipsoids and using the John basis, we also obtained the L_p-analogs of Loomis-Whitney inequality and the Pythagorean inequality.
引文
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