Fourier Analysis in Convex Geometry

One of the results discussed in the book is Ball's theorem, establishing the exact upper bound for the  -dimensional volume of hyperplane sections of the  -dimensional unit cube (it is      for each  ). Another is the Busemann-Petty problem: if   and   are two convex origin-symmetric  -dimensional bodies and the  -dimensional volume of each central hyperplane section of   is less than the  -dimensional volume of the corresponding section of  , is it true that the  -dimensional volume of   is less than the volume of  ? (The answer is positive for   and negative for  .)

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.