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Intersection bodies and Lp spaces
 Adv. Math
"... Abstract. We prove that convex intersection bodies are isomorphically equivalent to unit balls of subspaces of Lq for each q ∈ (0, 1). This is done by extending to negative values of p the factorization theorem of Maurey and Nikishin which states that for any 0 < p < q < 1 every Banach subs ..."
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Abstract. We prove that convex intersection bodies are isomorphically equivalent to unit balls of subspaces of Lq for each q ∈ (0, 1). This is done by extending to negative values of p the factorization theorem of Maurey and Nikishin which states that for any 0 < p < q < 1 every Banach subspace of Lp is isomorphic to a subspace of Lq. 1.
Sections of star bodies and the Fourier transform
 Proceedings of the AMSIMSSIAM Summer Research Conference in Harmonic Analysis, Mt Holyoke, 2001, Contemp. Math
, 2003
"... A new approach to the study of sections of star bodies, based on methods of Fourier analysis, has recently been developed. The idea is to express certain geometric properties of bodies in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. This ap ..."
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A new approach to the study of sections of star bodies, based on methods of Fourier analysis, has recently been developed. The idea is to express certain geometric properties of bodies in terms of the Fourier transform and then apply methods of harmonic analysis to solve geometric problems. This approach has already led to several results including an analytic solution to the BusemannPetty problem on sections of convex bodies. In this article we bring these results together and present short proofs of major connections.
A note on positive definite norm dependent functions
 Proceedings of the Conference on High Dimensional Probability, Luminy
, 2008
"... Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f: [0, ∞) → R, that if the function f(‖x‖K) is positive definite on R n, then the space (R n, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and ..."
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Abstract. Let K be an origin symmetric star body in R n. We prove, under very mild conditions on the function f: [0, ∞) → R, that if the function f(‖x‖K) is positive definite on R n, then the space (R n, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of ndimensional versions, i.e. random vectors X = (X1,...,Xn) in R n such that the random variables ∑ aiXi are identically distributed for all a ∈ R n, up to a constant depending on ‖a‖K only. 1.
POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES
, 903
"... Abstract. We say that a random vector X = (X1,..., Xn) in R n is an ndimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions ..."
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Abstract. We say that a random vector X = (X1,..., Xn) in R n is an ndimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an ndimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L0. This result is almost optimal, as the norm of any finite dimensional subspace of Lp with p ∈ (0, 2] is the standard of an ndimensional version (pstable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form f( ‖ · ‖K) is positive definite on R n, where K is an origin symmetric star body in R n and f: R → R is an even continuous function, then either the space (R n, ‖ · ‖K) embeds in L0 or f is a constant function. Combined with known facts about embedding in L0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions. 1.