本文選題：log-Minkowski不等式 + Minkowski不等式��； 參考：《西南大學(xué)》2017年博士論文

【摘要】：經(jīng)典的 Brunn-Minkowski 不等式與 Minkowski 不等式是 Brunn-Minkowski理論中最重要的幾何不等式,是經(jīng)典等周不等式的自然推廣.2012年,Boroczky-Lutwak-Yang-Zhang給出了平面中關(guān)于原點對稱的凸體的log-Minkowski不等式及 log-Brunn-Minkowski 不等式,并猜想 log-Minkowski 不等式及 log-Brunn-Minkowski不等式對高維空間中原點對稱的凸體也成立.猜想的log-Minkowski不等式及l(fā)og-Brunn-Minkowski不等式加強了經(jīng)典的Brunn-Minkowski不等式與Minkowski不等式,且在解決log-Minkowski問題唯一性中至關(guān)重要.目前高維空間中關(guān)于原點對稱凸體的log-Minkowski不等式的研究較多,而非對稱凸體的log-Minkowski不等式的研究困難重重.最近,A.Stancu研究了高維空間中非對稱凸體的log-Minkowski不等式,并證明了高維空間中特殊情形下猜想的log-Minkowski 不等式.受Boroczky-Lutwak-Yang-Zhang和A.Stancu研究的啟發(fā),本學(xué)位論文著重研究空間中非對稱凸體的log-Minkowski不等式.第二章中首先介紹了平面中已知的log-Minkowski不等式和log-Brunn-Minkowski不等式,然后給出高維空間中非對稱凸體的log-Minkowski不等式及l(fā)og-Minkowski型不等式(猜想的log-Minkowski不等式的等價形式),這些不等式推廣了 A.Stancu的結(jié)果.第三章探討了對偶的log-Minkowski不等式及其等價形式.在最后一章中,我們得到一個關(guān)于p-仿射表面積的不等式,它是p—仿射等周不等式的自然推廣.同時,我們還給出了 Mahler猜想的一個近似估計,即凸體與其極體體積之積(仿射不變量)的下界估計。
[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.
【學(xué)位授予單位】：西南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O186.5

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1 王星星;關(guān)于非對稱凸體的log-Minkowski不等式[D];西南大學(xué);2017年

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