凸体及星体的不等式与极值问题
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摘要
本文首先介绍凸体几何的发展历史和各主要研究方向的发展概况，本博士论文以研究一般凸体、星体以及单形和超平行体等特殊类凸体的度量不等式和极值问题为主要内容，研究工作分为两个方面。
一方面是利用几何分析的渐近理论、局部理论和积分变换方法研究一般凸体和星体的度量不等式和极值问题，由第二章和第三章构成。由于Petty-Schneider问题是凸体几何中一个热点问题，第二章首先推广Petty-Schneider问题到一般的均质积分情形；Ball在研究Petty-Schneider问题时讨论了球和立方体的截面性质，受Ball思想的启发，我们给出了球的截面的两个新的度量不等式；关于凸体的Brunn-Minkowski不等式是凸体理论的精髓，混合投影体的Brunn-Minkowski不等式也由Lutwak所证明，我们则证明了投影体的极体的Brunn-Minkowski型不等式。星体的对偶Brunn-Minkowski理论是上世纪70年代产生的新兴研究领域，第三章我们建立了星体对偶均质积分的两个新型不等式，它们形式上类似于正实数的初等对称函数的Marcus-Lopes不等式和Bergstrom不等式，也类似于行列式的Fan Ky不等式。此外，我们还证明了星体的对偶混合均质积分和对偶混合p-均质积分的相关性质。
另一方面的工作是利用外微分和代数的方法研究一些特殊类的凸体(如单形、超平行体等)的度量不等式和极值性质，这方面的研究工作由第四章、第五章和第六章构成。凸体的混合体积为几何中的各类度量提供了统一的处理模式，它是有限个凸体的连续函数，本文第四章引入凸体混合体积的离散形式：两个有限向量集的混合体积的概念，同时利用外微分为工具证明了向量集的混合体积与由两向量集分别张成的平行体体积之间的一个强有力的不等式；Cayley-Menger行列式是解决有限点集不等式和嵌入问题的极好工具，我们则定义了两个点集的混合Cayley-Menger行列式，获得了混合Cayley-Menger行列式与向量集的混合体积以及两个单形体积乘积之间的关系，这个关系式容量大，包含了不少的经典度量关系和近期被发现的新结果；我们还引入两个单形的混合距离矩阵的概念，证明了它的行列式与两单形的外径的等量关系。在第四章最后，利用我们获得的主要结论简洁地证明了如单形的正弦定理和平行体的Hadamaxd不等式的逆形式等一些著名的结论。第五章的任务是利用一个分析不等式和杨路-张景中质点组不等式和权变换的方法把关于两个三角形的Klamkin不等式推广到高维空间，同时建立一系列的涉及单形的体积、各面面积、任意点到单形的各顶点距离的新的不等式。第六章我们利用外微分的方
II
法，首次给出了。维单形中面的解析表达式，并且证明了单形中面类似于三角形中
线的性质，如对于一个给定的单形，存在另一个单形使得它的棱长分别等于给定单
形的中面面积的值，一个单形的所有中面有且仅有一个公共点等.进一步，利用中
面的表达式建立了一系列涉及单形的棱长、各面面积、外径、中线长等的新的不等
式。
The development survey and main research directions of convex geometry are presented in the preface. This Ph.D. dissertation research the inequalities and extremum properties for convex bodies, star bodies and some specific convex bodies such as simpli-cies and parallelotopes. The research works of this thesis consists of two parts.
In the first aspect, some inequalities and extremum properties are established by applying the asymptotic theory, local theory and integral transforms. The Petty-Schneider problem has attracted the attention of those working in convex geometry. Chapter 2 extend Petty-Schneider theorem for projection bodies (zonoids) to quermassintegrals. Two inequalities for sections of centered bodies are given, which are motivated by the so-called hyperplane conjecture for convex bodies. The classical Brunn-Minkowski's inequality is the heart of convex bodies theory. The Brunn-Minkowski's inequality for mixed projection bodies was obtained by Lutwak. We find the Brunn-Minkowski's inequality for the polar of mixed projection bodies. The dual Brunn-Minkowski theory earns its place as an essential tool in geometric tomography. In Chapter 3, the inequalities about the dual quermassintegrals of star bodies in Rn are established, which are analogue not only to Marcus-Lopes's inequality and Bergstrom's inequality for elementary s
ymmetric functions of positive reals, but also to Fan Ky's inequality for determinant. On the other hand, the dual mixed Quermassintegrals and the dual mixed p- Quermassintegrals are introduced. We generalize the dual Brunn-Minkowski Theory.
The other part of the research work is presented in Chapter 4, Chapter 5 and Chapter 6. We find the inequalities and extremum properties for specific convex bodies such as simplicies and parallelotopes by employing the exterior differential methods and algebraic means. The theory of mixed volumes provides a unified treatment of various important metric quantities in geometry, such as volume, surface area and mean width. Chapter 4 introduce the concept of the mixed volume of two finite vector sets in Rn, which can be regard as the discrete form of mixed volume of two convex bodies. An new and powerful inequality associating with the mixed volume of two finite vector sets is obtained. The Cayley-Menger determinant has proved extraordinarily useful tool in dealing with some inequalities for finite points sets. We introduce the mixed Cayley-Menger determinant
and obtain the formula for volume product of two simplices which contains a lot of papers on the properties of a pair of triangles (simplices). Meanwhile, the relation concerning the determinant of mixed distance matrix and the circum-radius of two n-simplices is given. Besides, employing new and simple methods, some well-known results of simplices and Hadamard inequality are reproved. In Chapter 5, by applying an analytic inequality and the moment of inertia inequality in Rn, we generalize the Klamkin's inequality to several dimensions and establish some inequalities for the volume, facet areas and distances between any point of Rn and vertices of an n-simplex. To obtain the analytic expression for the mid-facet area of a n-dimensional simplex, the exterior differential method is applied in Chapter 6. Furthermore, some properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex, there exists another simplex such that its edge-lengths equal to th
ese mid-facets area respectively, and all of the mid-facets of a simplex have a common point) are confirmed. In the end , by using the analytic expression, a number of inequalities which combine edge-lengths, circum-radius and median line with the mid-facet area for a simplex are established.
一方面是利用几何分析的渐近理论、局部理论和积分变换方法研究一般凸体和星体的度量不等式和极值问题，由第二章和第三章构成。由于Petty-Schneider问题是凸体几何中一个热点问题，第二章首先推广Petty-Schneider问题到一般的均质积分情形；Ball在研究Petty-Schneider问题时讨论了球和立方体的截面性质，受Ball思想的启发，我们给出了球的截面的两个新的度量不等式；关于凸体的Brunn-Minkowski不等式是凸体理论的精髓，混合投影体的Brunn-Minkowski不等式也由Lutwak所证明，我们则证明了投影体的极体的Brunn-Minkowski型不等式。星体的对偶Brunn-Minkowski理论是上世纪70年代产生的新兴研究领域，第三章我们建立了星体对偶均质积分的两个新型不等式，它们形式上类似于正实数的初等对称函数的Marcus-Lopes不等式和Bergstrom不等式，也类似于行列式的Fan Ky不等式。此外，我们还证明了星体的对偶混合均质积分和对偶混合p-均质积分的相关性质。
另一方面的工作是利用外微分和代数的方法研究一些特殊类的凸体(如单形、超平行体等)的度量不等式和极值性质，这方面的研究工作由第四章、第五章和第六章构成。凸体的混合体积为几何中的各类度量提供了统一的处理模式，它是有限个凸体的连续函数，本文第四章引入凸体混合体积的离散形式：两个有限向量集的混合体积的概念，同时利用外微分为工具证明了向量集的混合体积与由两向量集分别张成的平行体体积之间的一个强有力的不等式；Cayley-Menger行列式是解决有限点集不等式和嵌入问题的极好工具，我们则定义了两个点集的混合Cayley-Menger行列式，获得了混合Cayley-Menger行列式与向量集的混合体积以及两个单形体积乘积之间的关系，这个关系式容量大，包含了不少的经典度量关系和近期被发现的新结果；我们还引入两个单形的混合距离矩阵的概念，证明了它的行列式与两单形的外径的等量关系。在第四章最后，利用我们获得的主要结论简洁地证明了如单形的正弦定理和平行体的Hadamaxd不等式的逆形式等一些著名的结论。第五章的任务是利用一个分析不等式和杨路-张景中质点组不等式和权变换的方法把关于两个三角形的Klamkin不等式推广到高维空间，同时建立一系列的涉及单形的体积、各面面积、任意点到单形的各顶点距离的新的不等式。第六章我们利用外微分的方
II
法，首次给出了。维单形中面的解析表达式，并且证明了单形中面类似于三角形中
线的性质，如对于一个给定的单形，存在另一个单形使得它的棱长分别等于给定单
形的中面面积的值，一个单形的所有中面有且仅有一个公共点等.进一步，利用中
面的表达式建立了一系列涉及单形的棱长、各面面积、外径、中线长等的新的不等
式。
The development survey and main research directions of convex geometry are presented in the preface. This Ph.D. dissertation research the inequalities and extremum properties for convex bodies, star bodies and some specific convex bodies such as simpli-cies and parallelotopes. The research works of this thesis consists of two parts.
In the first aspect, some inequalities and extremum properties are established by applying the asymptotic theory, local theory and integral transforms. The Petty-Schneider problem has attracted the attention of those working in convex geometry. Chapter 2 extend Petty-Schneider theorem for projection bodies (zonoids) to quermassintegrals. Two inequalities for sections of centered bodies are given, which are motivated by the so-called hyperplane conjecture for convex bodies. The classical Brunn-Minkowski's inequality is the heart of convex bodies theory. The Brunn-Minkowski's inequality for mixed projection bodies was obtained by Lutwak. We find the Brunn-Minkowski's inequality for the polar of mixed projection bodies. The dual Brunn-Minkowski theory earns its place as an essential tool in geometric tomography. In Chapter 3, the inequalities about the dual quermassintegrals of star bodies in Rn are established, which are analogue not only to Marcus-Lopes's inequality and Bergstrom's inequality for elementary s
ymmetric functions of positive reals, but also to Fan Ky's inequality for determinant. On the other hand, the dual mixed Quermassintegrals and the dual mixed p- Quermassintegrals are introduced. We generalize the dual Brunn-Minkowski Theory.
The other part of the research work is presented in Chapter 4, Chapter 5 and Chapter 6. We find the inequalities and extremum properties for specific convex bodies such as simplicies and parallelotopes by employing the exterior differential methods and algebraic means. The theory of mixed volumes provides a unified treatment of various important metric quantities in geometry, such as volume, surface area and mean width. Chapter 4 introduce the concept of the mixed volume of two finite vector sets in Rn, which can be regard as the discrete form of mixed volume of two convex bodies. An new and powerful inequality associating with the mixed volume of two finite vector sets is obtained. The Cayley-Menger determinant has proved extraordinarily useful tool in dealing with some inequalities for finite points sets. We introduce the mixed Cayley-Menger determinant
and obtain the formula for volume product of two simplices which contains a lot of papers on the properties of a pair of triangles (simplices). Meanwhile, the relation concerning the determinant of mixed distance matrix and the circum-radius of two n-simplices is given. Besides, employing new and simple methods, some well-known results of simplices and Hadamard inequality are reproved. In Chapter 5, by applying an analytic inequality and the moment of inertia inequality in Rn, we generalize the Klamkin's inequality to several dimensions and establish some inequalities for the volume, facet areas and distances between any point of Rn and vertices of an n-simplex. To obtain the analytic expression for the mid-facet area of a n-dimensional simplex, the exterior differential method is applied in Chapter 6. Furthermore, some properties of the mid-facets of a simplex analogous to median lines of a triangle (such as for all mid-facets of a simplex, there exists another simplex such that its edge-lengths equal to th
ese mid-facets area respectively, and all of the mid-facets of a simplex have a common point) are confirmed. In the end , by using the analytic expression, a number of inequalities which combine edge-lengths, circum-radius and median line with the mid-facet area for a simplex are established.
引文
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[127] 杨路，张景中，关于有限点集的一类几何不等式，数学学报，23，5(1980)，740-749．
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[129] 张景中，杨路，关于质点组的一类几何不等式，中国科技大学学报，11，2(1981)，1-8．
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[132] Zhang G. Y., Dual Kinematic formulas, Trans. Amer. Math. Soc., 351(1999), 985-995.
[133] Zhang G. Y., A positive answer in the Busemann Petyy roblem in R~4, Ann. Math., 149(1999), 535-543.
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[135] 张晗方，《几何不等式导引》，中国科学文化出版社，(2003)
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