关于非对称凸体的log-Minkowski不等式

发布时间：2018-07-03 09:24

本文选题：log-Minkowski不等式 + Minkowski不等式　；参考：《西南大学》2017年博士论文

【摘要】：经典的 Brunn-Minkowski 不等式与 Minkowski 不等式是 Brunn-Minkowski理论中最重要的几何不等式,是经典等周不等式的自然推广.2012年,Boroczky-Lutwak-Yang-Zhang给出了平面中关于原点对称的凸体的log-Minkowski不等式及 log-Brunn-Minkowski 不等式,并猜想 log-Minkowski 不等式及 log-Brunn-Minkowski不等式对高维空间中原点对称的凸体也成立.猜想的log-Minkowski不等式及log-Brunn-Minkowski不等式加强了经典的Brunn-Minkowski不等式与Minkowski不等式,且在解决log-Minkowski问题唯一性中至关重要.目前高维空间中关于原点对称凸体的log-Minkowski不等式的研究较多,而非对称凸体的log-Minkowski不等式的研究困难重重.最近,A.Stancu研究了高维空间中非对称凸体的log-Minkowski不等式,并证明了高维空间中特殊情形下猜想的log-Minkowski 不等式.受Boroczky-Lutwak-Yang-Zhang和A.Stancu研究的启发,本学位论文着重研究空间中非对称凸体的log-Minkowski不等式.第二章中首先介绍了平面中已知的log-Minkowski不等式和log-Brunn-Minkowski不等式,然后给出高维空间中非对称凸体的log-Minkowski不等式及log-Minkowski型不等式(猜想的log-Minkowski不等式的等价形式),这些不等式推广了 A.Stancu的结果.第三章探讨了对偶的log-Minkowski不等式及其等价形式.在最后一章中,我们得到一个关于p-仿射表面积的不等式,它是p—仿射等周不等式的自然推广.同时,我们还给出了 Mahler猜想的一个近似估计,即凸体与其极体体积之积(仿射不变量)的下界估计。
[Abstract]:The classical Brunn-Minkowski inequality and Minkowski inequality are the most important geometric inequalities in Brunn-Minkowski theory and a natural generalization of classical isoperimetric inequalities. Boroczky-Lutwak-Yang-Zhang gave log-Minkowski inequality and log-Brunn-Minkowski inequality for convex bodies with symmetric origin in 2012. It is conjectured that log-Minkowski inequality and log-Brunn-Minkowski inequality also hold for convex bodies with symmetric origin in high dimensional space. The conjecture log-Minkowski inequality and log-Brunn-Minkowski inequality strengthen the classical Brunn-Minkowski inequality and Minkowski inequality, and are very important in solving the uniqueness of log-Minkowski problem. At present, there are many researches on log-Minkowski inequality of origin symmetric convex body in high-dimensional space, but it is difficult to study log-Minkowski inequality for asymmetric convex body. Recently, A. Stancu studied the log-Minkowski inequality for asymmetric convex bodies in higher dimensional spaces, and proved the log-Minkowski inequality for conjecture in special cases in higher dimensional spaces. Inspired by Boroczky-Lutwak-Yang-Zhang and A. Stancu, this dissertation focuses on the log-Minkowski inequality for asymmetric convex bodies in space. In the second chapter, the known log-Brunn-Minkowski inequality and log-Brunn-Minkowski inequality are introduced. Then the log-Minkowski inequality and log-Minkowski type inequality (the equivalent form of the conjecture log-Minkowski inequality) for asymmetric convex bodies in higher dimensional space are given. These inequalities generalize the results of A. Stancu. In chapter 3, we discuss the dual log-Minkowski inequality and its equivalent form. In the last chapter, we obtain an inequality about p-affine surface area, which is a natural generalization of p-affine isoperimetric inequality. At the same time, we give an approximate estimate of Mahler's conjecture, that is, the lower bound estimate of the product (affine invariant) of the volume of convex body and its polar body.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O186.5

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