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We say that an even continuous function H on the unit sphere Ω in Rn admits the Blaschke-Levy representation with q > 0 if there exists an even function b ∈ L1(Ω) so that H q(x) = ∫ Ω |(x, ξ)|qb(ξ) dξ for every x ∈ Ω. This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple… (More)

- N J Kalton, A Koldobsky
- 2002

For 0 < p < 1 we give examples of Banach spaces isometrically embedding into Lp but not into any Lr with p < r ≤ 1.

A comparison problem for volumes of convex bodies asks whether inequalities fK(ξ) ≤ fL(ξ) for all ξ ∈ S imply that Voln(K) ≤ Voln(L), where K,L are convex bodies in R, and fK is a certain geometric characteristic of K. By linear stability in comparison problems we mean that there exists a constant c such that for every ε > 0, the inequalities fK(ξ) ≤ fL(ξ)… (More)

The Busemann-Petty problem for an arbitrary measure μ with non-negative even continuous density in R asks whether origin-symmetric convex bodies in R with smaller (n − 1)-dimensional measure μ of all central hyperplane sections necessarily have smaller measure μ. It was shown in [Zv] that the answer to this problem is affirmative for n ≤ 4 and negative for… (More)

- Paul D. Gader, Mohamed A. Khabou, Alexander Koldobsky
- Pattern Recognition
- 2000

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.… (More)

We define embedding of an n-dimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lp-spaces. The well-known connection between embeddings into Lp and positive definite functions is extended to the case of negative p by showing that a normed space embeds in L−p if and only if ‖x‖… (More)

For every n ≥ 3, we construct an n-dimensional Banach space which is isometric to a subspace of L1/2 but is not isometric to a subspace of L1. The isomorphic version of this problem (posed by S. Kwapien in 1969) is still open. Another example gives a Banach subspace of L1/4 which does not embed isometrically in L1/2. Note that, from the isomorphic point of… (More)

The Busemann-Petty problem asks whether origin-symmetric convex bodies inR with smaller areas of all central hyperplane sections necessarily have smaller n-dimensional volume. The solution was completed in the end of the 90’s, and the answer is affirmative if n ≤ 4 and negative if n ≥ 5. Since the answer is negative in most dimensions, it is natural to ask… (More)