Applications of Fourier analysis to convex geometry

摘要：PI计划使用调和分析和泛函分析的方法研究凸体的几何形状。一个方向是用傅里叶变换来表示凸体的不同性质，然后用傅里叶分析的方法来求解几何问题。这种方法将导致关于凸体的截面和投影的新结果，不同类别物体的特征，Busemann-Petty类型的新结果。另一个研究方向与凸几何和l_p -空间理论之间的新联系有关，该联系最近被pi发现，并允许通过将关于l_p -空间的不同事实推广到p的负值来获得新的几何结果。特别有趣的是相交体和偏振投影体是否同构等价的问题。这个问题的解决将代表着理解截面和投影之间的对偶性的一大步，这仍然是凸几何中最有趣的谜团之一。本文还计划将这些方法应用于几个渐近几何分析的开放问题，包括凸体的中心极限问题。基于凸体的截面和投影信息来研究凸体的几何性质在数学和科学的许多领域都有重要的应用，从x射线和几何断层扫描的经典理论到工程和医学的不同问题。PI最近发展了一种基于傅立叶分析方法的凸体截面和投影的新方法。这种方法已经导致了几个结果，包括一个关于凸体截面的Busemann-Petty问题的傅立叶解析解，该问题询问中心截面均匀较小的物体是否必然具有较小的体积。所提出的研究将有助于更好地理解凸体的几何形状，并将进一步将凸体几何的方法和结果与调和分析、泛函分析和概率分析联系起来。在这个项目中开发的计算傅里叶变换的新技术将具有独立的价值，并可应用于信号处理和统计。

Abstract: The PI plans to study the geometry of convex bodies usingmethods of harmonic and functional analysis. One directionis to express different properties of convex bodies in termsof the Fourier transform and then use methods of Fourieranalysis to solve geometric problems. This approach will leadto new results on sections and projections of convex bodies,characterizations of different classes of bodies, new resultsof the Busemann-Petty type. Another direction of research isrelated to a new connection between convex geometry and thetheory of L_p-spaces, which has recently been found by the PIand allows to get new geometric results by extending differentfacts about L_p-spaces to negative values of p. Particularlyinteresting is the question of whether intersection and polarprojection bodies are isomorphically equivalent. A solution tothis problem will represent a big step in understanding theduality between sections and projections, which remains oneof the most intriguing mysteries of convex geometry. The PIalso plans to apply these methods to several open problems ofasymptotic geometric analysis, including the central limitproblem for convex bodies.The study of geometric properties of convex bodies based oninformation about sections and projections of these bodies hasimportant applications to many areas of mathematics and science,from the classical theory of x-rays and geometric tomography todifferent problems of engineering and medicine. A new approach tosections and projections of convex bodies, based on methods ofFourier analysis, has recently been developed by the PI. Thisapproach has already led to several results, including a Fourieranalytic solution of the Busemann-Petty problem on sections ofconvex bodies asking whether bodies with uniformly smaller centralsections necessarily have smaller volume. The proposed researchwill lead to better understanding of the geometry of convex bodiesand will further relate methods and results of convex geometry toharmonic analysis, functional analysis and probability. New techniquesfor computing the Fourier transform, developed in this project, willhave independent value and can be applied to signal processing andstatistics.