A. Koldobsky

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We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n 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(More)
The Busemann-Petty problem asks whether origin-symmetric convex bodies in R n with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer is affirmative if n ≤ 4 and negative if n ≥ 5. In this article we modify the assumptions of the original Busemann-Petty problem to guarantee the affirmative answer(More)
We present a formula for the Fourier transforms of order statistics in R n showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in R n. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces into L q [10] to prove that, for n ≥ 3 and q ≤ 1, the(More)
Suppose that we have the unit Euclidean ball in R n and construct new bodies using three operations-linear transformations, closure in the radial metric and multiplicative summation defined by xK+ 0 L = xK xL. We prove that in dimension 3 this procedure gives all origin symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We(More)
We prove that convex intersection bodies are isomor-phically equivalent to unit balls of subspaces of L q for each q ∈ (0, 1). This is done by extending to negative values of p the fac-torization theorem of Maurey and Nikishin which states that for any 0 < p < q < 1 every Banach subspace of L p is isomorphic to a subspace of L q .
Suppose that we have the unit Euclidean ball in R n and construct new bodies using three operations-linear transformations, closure in the radial metric and multiplicative summation defined by xK+ 0 L = p xK xL. We prove that in dimension 3 this procedure gives all origin symmetric convex bodies, while this is no longer true in dimensions 4 and higher. We(More)