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)
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C n with smaller central hyper-plane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4.
For 0 < p < 1 we give examples of Banach spaces isometrically embedding into L p but not into any L r with p < r ≤ 1.
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  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 point out a certain class of functions f and g for which random) are non-negatively correlated for any symmetric jointly stable random variables X i. We also show another result that is related to the correlation problem for Gaussian measures of symmetric convex sets.
We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann's theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex… (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)