An analytic solution to the Busemann-Petty problem on sections of convex bodies

R.J. Gardner, A. Koldobsky and T. Schlumprecht

Review from Zentralblatt MATH:

Let $K$ be an origin-symmetric star body in $\Bbb R^n$ with $C^\infty$ boundary, and let $k\in\Bbb N \cup\{0\}, k\neq n - 1.$ Suppose that $\xi\in S^{n-1},$ and let $A_\xi$ be the corresponding parallel section function of $K.$ The function $A_\xi$ (or $(n - 1)$-dimensional $X$-ray) gives the ($(n - 1)$-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction.

The authors derive a formula connecting the derivatives of $A_\xi$ with the Fourier transform (in the sense of distributions) of powers $(\rho_K^{n-k-1})^\wedge$ of the radial function $\rho_K$ of the body: $$(\rho_K^{n-k-1})^\wedge(\xi) =\cases (-1)^{k/2}\pi(n - k - 1)A_\xi^{(k)}(0),&\text{if $k$ is even,}
c_k\int_0^\infty\frac{A_\xi(z)-A_\xi(0)- A_\xi''(0)\frac{z^2}{2}-\dots-A_\xi^{(k-1)}(0)\frac{z^{k-1}}{(k-1)! }} {z^{k+1}} dz,& \text{if $k$ is odd,} \endcases $$ where $c_k=(-1)^{(k+1)/2}2(n-1-k)k!.$ This formula provides a new characterization of intersection bodies in $\Bbb R^n$ and leads to a unified analytic solution to the Busemann-Petty problem: If the section function of a centered convex body in $\Bbb R^n$, $n\ge 3,$ is smaller than that of another such body, is its volume also smaller? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, this formula shows that the answer to the problem depends on the behavior of the $(n - 2)$-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for $n \le 4$ and the negative answer for $n\ge 5$ now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

Reviewed by Serguey M.Pokas

Keywords: convex body; star body; Busemann-Petty problem; intersection body; Fourier transform; Radon transform; convexity; parallel section

Classification (MSC2000): 52A20 46B07 42B10

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