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Full-Text Articles in Physical Sciences and Mathematics
An Analytic Solution To The Busemann-Petty Problem On Sections Of Convex Bodies, Richard J. Gardner, Alexander Koldobsky, Thomas Schlumprecht
An Analytic Solution To The Busemann-Petty Problem On Sections Of Convex Bodies, Richard J. Gardner, Alexander Koldobsky, Thomas Schlumprecht
Mathematics Faculty Publications
We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in Rn with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n - 1)-dimensional X-ray) gives the ((n - 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in Rn and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in Rn such that the …
On The Busemann-Petty Problem Concerning Central Sections Of Centrally Symmetric Convex-Bodies, Richard J. Gardner
On The Busemann-Petty Problem Concerning Central Sections Of Centrally Symmetric Convex-Bodies, Richard J. Gardner
Mathematics Faculty Publications
We present a method which shows that in E3 the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.
Intersection Bodies And The Busemann-Petty Problem, Richard J. Gardner
Intersection Bodies And The Busemann-Petty Problem, Richard J. Gardner
Mathematics Faculty Publications
It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d ≤ 4, and that suitably smooth axis-convex bodies of revolution are intersection bodies when d ≤ 4. These results show that the Busemann-Petty problem has a negative answer for d ≥ 5 …