基于凸体的截面和投影信息来研究其几何性质，在数学和科学的诸多领域都有着重要应用。本书探讨了一种新的傅里叶分析方法，其核心思路是将凸体的某些几何性质通过傅里叶分析进行表述，并借助调和分析方法解决几何问题。One of the results discussed in the book is Ball's theorem, establishing the exact upper bound for the (n?1)-dimensional volume of hyperplane sections of the n-dimensional unit cube (it is 2 for each n≥2). Another is the Busemann–Petty problem: if K and L are two convex origin-symmetric n-dimensional bodies and the (n?1)-dimensional volume of each central hyperplane section of K is less than the (n?1)-dimensional volume of the corresponding section of L, is it true that the n-dimensional volume of K is less than the volume of L? (The answer is positive for n≤4 and negative for n>4.)The book is suitable for graduate students and researchers interested in geometry, harmonic and functional analysis, and probability. Prerequisites for reading this book include basic real, complex, and functional analysis.