【作者】 朱先阳；

【导师】 冷岗松；

【作者基本信息】 上海大学 ， 运筹学与控制论， 2008， 博士

【摘要】 本学位论文致力于研究L_p-空间中凸体几何的度量不等式和极值问题,隶属于L_p-Brunn-Minkowski理论（又称为Brunn-Minkowski-Firey理论）领域,该领域是近十多年来在国际上发展非常迅速的一个几何学分支。本文主要利用L_p-Brunn-Minkowski理论的基本概念、基本知识和积分变换方法,研究了L_p-空间中凸体几何领域诸多几何体:L₂-投影体,L_p-截面体,混合截面体,混合新几何体Γ_-p,iK以及由我们新引入的L_p-仿射表面积、L_p-混合均质积分、L_p-混合仿射表面积所构成的度量不等式和极值问题。利用∏₂K是一个中心在原点的椭球这一特殊性质,在第二章给出了投影不等式Petty猜想的L_p-形式（p=1时即是Petty投影不等式猜想本身）在当p=2时,其逆形式的两个结果,这些结果是对投影不等式Petty猜想的完善,同时建立了L₂-投影体∏₂K与标准化下经典投影体∏K的包含关系,还解答了一个约束最小化问题。在第三章中,我们依据Gardner和Giannopoulos,V.Yaskin和M.Yaskin,C.Haberel和M.Ludwig提出的概念,合理地用不同的符号重新定义了L_p-截面体I_pK。对这一几何体,给出了算子I_p的线性等价性,及其当p≤-1时的单调性结果,同时将L_p-截面体和L_p-对偶混合均质积分（?）_-p,i（K,L）相结合,在正规化L_p-径向加或者L_p-径向线性组合的条件下,分别建立了关于L_p-截面体的对偶均质积分形式的Brunn-Minkowski不等式及其隔离加强形式。Lutwak推广了Winterniz单调性问题,即得到:假设K∈Fⁿ,且E是一个椭球,以及K的投影体的体积不超过E的投影体的体积,那么Ω（K）≤Ω（E）成立。在第四章,利用在L_p-混合体积和L_p-广义仿射表面积之间所建立的等价关系,我们把上述Lutwak的结果推广到L_p-形式,而且作为这种等价关系的应用,建立了L_p-混合均质积分形式的Aleksandrov的投影定理和Petty-Schneider定理。在凸几何中,凸体的极体是一个非常重要的研究对象。在最重要的仿射等周不等式中,Blaschke-Santaló不等式和Petty投影不等式是两个与凸体的极体密切相关的不等式,可是对于星体而言,其极体却并不一定存在。在第五章,利用Moszy（?）ska介绍的星对偶概念,将混合截面体的星对偶与对偶均质积分结合起来,在调和p-组合或者p-径向线性组合的条件下,分别建立了关于混合截面体的星对偶的对偶均质积分形式的Brunn-Minkowski不等式。根据Lutwak,Yang和Zhang提出的新几何体Γ_-pK,我们在第六章引入混合新几何体Γ_-p,iK的概念—新几何体是它的特殊情形。对混合新几何体Γ_-p,iK,我们获得了算子Γ_-p,i的五个基本性质,建立了Γ_-p,iK的体积V(Γ_-p,iK)与凸体K的体积V（K）之间的关系,以及关于混合新几何体Γ_-p,iK的Shephard类问题。此外,我们还在第七章研究了L_p-混合仿射表面积和L_p-质心体的相互关系,获得了Busemann-Petty仿射质心不等式的广义L_p-形式,得到了关于L_p-混合仿射表面积的Blaschke-Santaló不等式,同时,获得了对偶Urysohn不等式的一般形式,最后,利用p-Blaschke平行体的定义,我们建立了与Marcus-Lopes,Bergstrom和Ky Fan不等式的一种类似形式。更多还原

【Abstract】 The thesis is devoted to the study of metric inequalities and extremum problems in convex bodies geometry of L_p-space,and belongs to the domain,which is a high-speed developing geometry branch on the decade of late,of the L_p Brunn-Minkowski theory（or called Brunn-Minkowski-Firey theory）.By applying the basic notions,basic theories and integral transforms of the L_p Brunn-Minkowski theory,we reseach the metric inequalities and extremum properties of some geometry bodies containing the L₂-projection body, L_p-intersection body,mixed intersection bodies,mixed new geometry bodyΓ_p,iK and a new notion from our definition—L_p—affine surface area,L_p-mixed quermassintegrals and L_p-mixed affine surface areas in the L_p-Brunn-Minkowski theory.In Chapter two we use the special property thatΠ₂K is an origin-centered ellipsoid to give two versions for the reverses of the L₂-Petty projection inequality（L₁-Petty projection inequality is just the Petty’s conjectured projection inequality）,at the samr time, the inclusive relationships between L₂-projection bodyΠ₂K and the classical projection bodyΠK are established,a contrained minimization problem is solved.In Chapter three,associated with the notions of Gardner’s and Giannoponlos’,or V.Yaskin’s and M.yaskin’s,or C.haberl’s and M.Ludwig’s,we reasonablly rewrite it usng some different signs—L_p-intersection body I_pK.On this geometry body,linearly equivalent property and several monotonicity results when p≤-1 are given,and together L_p-intersection body with Lp dual mixed quermassiintegrals（?）_-p,i（K,L）,under the normalized L_p radial addition and L_p radial linear combination,we respectively show the dual quermassintegrals version’s Brunn-Minkowski inequality and its isolate forms about L_p-intersection body.Lutwak extended Winterniz monotonicity problem,that is,if K∈Kⁿ and E is an ellipsoid,then if the areas of the projections of K do not exceed those of E,it follows thatΩ（K）≤Ω（E）.Chapter four use an equivalent relationship between L_p-mixed volumes and L_p-extended affine surface areas,we extend Lutwak’s result to L_p analog. As applications of this approach,we establish the L_p-mixed quermassintegrat version’s Aleksandrov’s projection theorem and Petty-Schneider theorem.In the context of convex geometry,the polar of a convex body is an important object, however,the polar of a star body may not exist.In Chapter five,by the notion of star dual of a star body that Moszynska introduced,and associated with star dual of mixed intersection bodies and dual quermassintegrals,with the harmonic p-combination and p-radial linear combination,we respectively state the dual quermassintegrals version’s Brunn-Minkowski inequality about star dual of mixed intersection bodies.Recently,Lutwak,Yang and Zhang posed the notion of new geometric bodyΓ_-pK, in Chapter six,we introduce a new notion—the mixed new geometry bodyΓ_-p,iK（then new geometric body being its a special case）.For this geometric body,we obtain five properties of the opertorΓ_-p,i,establish the relationship between the volumes ofΓ_-p,iK and that of convex body K,and study Shephard version’s problem about the mixed new geometric body.As an aside,in Chapter seven,the concepts of the ith L_p-mixed anne surface area and L_p-polar curvature images are introduced,together L_p-centroid body with p-Blaschke linear combination,we prove the generalization of L_p-versions of the Busemann-Petty affine centroid inequality,and get the inequality analogous to the Blaschke-Santalóinequality for the L_p-mixed anne surface area and the general forms of the dual Urysohn inequality,and obtain the similar to the inequality for Marcus-Lopes,Bergstrom and Ky Fan.更多还原