Applications Of Integral Transforms To Busemann-Petty-type Problems

The thesis is devoted to the study of applications of integral transforms to BusemannPetty-type problems. Nowadays intetral transforms are well used to solve a series of classic problems such as the celebrated Busemann-Petty probelem and its generalizations. At the same time, the lower dimensional generalized Busemann-Petty problem has attracted increased interest for this direction.In view of the strong power of the Radon transform and the Fourier transform of distribution used to solve a number of Busemann-Petty-type problems, we use these two transforms to study some unsolved Busemann-Petty-type problems.In Chapter two we use/-dimensional spherical Radon transform to establish an extension of Funk's section theorem. Our result has the following corollary: If K is a star body in R~n whose central /-slices have the same volume (with appropriate dimension) as the central/-slices of a centered body M, then the dual quermassintegrals satisfy (?)_j (M)≤(?)_j (K), for any 0≤j＜n - i, with equality if and only if K = M. The case g is a centered body implies Funk's section theorem.Chapter three is concerned with the generalized Busemann-Petty problem. The generalized Busemann-Petty problem asks whether origin-symmetric convex bodies in R~n with larger volume of all/-dimensional sections necessarily have larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i＞3. The problem is still open for i = 2, 3. We prove two specific affirmative answers to the generalized BusemannPetty problem if the body with smaller/-dimensional volume belongs to given classes. Our results generalize Zhang's specific affirmative answer to the generalized Busemann-Petty problem.In Chapter four we adopt a Fourier analytic approach to consider a monotonicity problem. For p＞0 Lutwak, Yang and Zhang introduced a star bodyГ_pK of a convex body K. We consider the question of whetherГ_pK CГ_pL implies vol(K)≤vol(L). Our results are dual forms for the studies ofГp by Lutwak in the case p = 1 and by Grinberg and Zhang in the case p＞1.Chapter five is devoted to the Lp-Winterniz monotonicity. The L_p-Winterniz monotonicity problem asks: If K and L are origin-symmetric convex bodies with positive con- tinuousρ-curvature functions in R~n, and the Firey projections of K is smaller than those of L, does it follow that theρ-affine surface area of K is smaller than the p-affine surface area of L? We show that the answer is positive if and only if n≤2 whenρ=1, and the answer is negative for anyρ＞1 and n≥2.In Chapter six we generalized Koldobsky's connections between an analytic generalization of the Busemann-Petty problem and the positive definite distributions. Our results show that the structure of the positive definite distributions is closely related to the analytic generalization of the Busemann-Petty problem which was posed by Koldobsky.