Applications of Fourier analysis to convex geometry

摘要：PI计划用调和和泛函分析的方法研究凸体的几何。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解决几何问题。这种方法将导致关于凸体的截面和投影的新结果、不同类物体的刻画、Busemann-Petty类型的新结果。另一个研究方向是关于凸几何和L_p-空间理论之间的一个新的联系，它是最近由PI发现的，它允许将关于L_p-空间的不同事实推广到p的负值，从而得到新的几何结果。特别有趣的是交和极投影体是否同构的问题。这个问题的解决将代表着在理解截面和投影之间的对偶性方面迈出了一大步，这仍然是凸几何中最有趣的谜团之一。PI还计划将这些方法应用于渐近几何分析的几个公开问题，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许多数学和科学领域都有重要的应用，从经典的X射线理论和几何层析到不同的工程和医学问题。PI最近发展了一种基于傅立叶分析方法的凸体旋转和投影的新方法。这种方法已经得到了几个结果，包括关于凸体截面上的Busemann-Petty问题的傅立叶解析解，该问题询问具有一致较小中心截面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，并将凸几何的方法和结果进一步与调和分析、泛函分析和概率论联系起来。本项目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.