摘要
本文隶属于Brunn-Minkowski理论领域，该领域是近几十年来在国际上发展非常迅速而重要的一个几何学分支．本学位论文利用几何分析中的凸体理论，积分变换方法和解析不等式理论，研究了凸体和星体的等周问题和相关的不等式问题．
本文的研究工作主要分为三个方面：
在经典Brunn-Minkowski理论中，我们推广了矩阵形式的Brunn-Minkowski不等式．作为应用，我们推广了Alexander度量加不等式．
在对偶Brunn-Minkowski理论中，我们引入了对偶调和均质积分概念，系统的研究了它的性质，并建立对偶调和均质积分的Brunn-Minkowski不等式，Blaschke-Santalo型不等式和Bieberbach不等式；接着我们建立了对偶仿射均质积分的对偶Brunn-Minkowski不等式，最近我们得知这个不等式被Gardner用另外的方式证明；凸体的极体是凸几何中一个重要概念，既然相交体和投影体有对偶关系，因此在研究完投影体的极体之后自然要研究相交体的极体．但相交体不一定凸，所以关于相交体的极体的性质和经典不等式的讨论与研究几乎是一片空白．采用Moszynska引入的星对偶的概念，我们在第五章中研究相交体的星对偶体，从而更进一步地揭示了投影体和相交体两者之间的对偶关系；引入了对偶混合体的概念．作为应用，我们给出了对偶Brunn-Minkowski不等式的加强版本．这些结果都加强了经典Brunn-Minkowski理论与对偶Brunn-Minkowski理论的对偶性．
在L_p-Brunn-Minkowski理论中，我们首先给出了两个关于新椭球的极值性质；然后研究了L_p仿射表面积，将Petty仿射投影不等式和Winterniz单调性问题推广到L_p仿射表面积；建立了L_p质心体和其极体的Brunn-Minkowski型不等式；研究了凸体的最小L_p平均宽度，给出凸体K具有最小L_p平均宽度的充分必要条件，最后给出了凸体L_p平均宽度位置的稳定性．
This article belong to the Brunn-Minkowski theory, which is a high-speed developing geometry branch during the past several decades. This thesis works for theoretical study on isoperimetric problem and related inequalities by using theory of geometry analysis, way of integral transformations and analysis inequalities.
The research works of this thesis consists of three parts.
In classical Brunn-Minkowski theory, we establish an extension of the matrix form of the Brunn-Minkowski inequality. As applications, we give generalizations on metric addition inequality of Alexander.
In dual Brunn-Minkowski theory, we study the properties of the dual harmonic quer-massintegrals systematically and establish some inequalities for the dual harmonic quer-massintegrals, such as the Minkowski inequality, the Brunn-Minkowski inequality, the Blaschke-Santalo inequality and the Bieberbach inequality. We establish the dual Brunn-Minkowski inequality for dual affine quermassintegrals. Recently we learned that Gardner have independently proved it by a different method. The polar body of a convex body is an important object in the context of convex geometry. Hence, after we studied the intersection bodies, it is natural to consider the inequalities for their polar bodies. But the intersection body of even a convex body generally is not convex. Thus the inequalities for the polar body of the intersection body can not be given in the general cases. Applying the concept of star dual, which is introduced by Moszynska, we establish some inequalities for star duals of intersection bodies. Hence, we present a new kind of duality between intersection bodies and projection bodies. At last, we introduce the dual mixed body and establish some properties and inequalities of it. As applications, we strength some inequalities in the the dual Brunn-Minkowski theory.
In Brunn-Minkowski-Firey theory, we establish two extremum properties of the new ellipsoid; Then we generalize Petty's affine projection inequality and monotonicity results related to affine surface area to L_p—affine surface area; We establish the Brunn-Minkowski type inequalities for the volume of the L_p centroid body and its polar body with respect to the normalized L_p radial addition. At last, we introduce the minimal L_p—mean width of a convex body and generalize the minimal mean width to the Brunn-Minkowski-Firey theory. Furthermore, we get the stability version of the L_p mean width position for L_p projection body.
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