Results 1  10
of
16
Projections Of Convex Bodies And The Fourier Transform
 OF CONVEX BODIES 11
"... The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the BusemannPetty problem, characterizations of intersection bodies, extremal sections of l p balls. In this article, we extend this appro ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the BusemannPetty problem, characterizations of intersection bodies, extremal sections of l p balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections of l p balls, and give a Fourier analytic solution to Shephard's problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function. 1.
Schoenberg’s problem on positive definite functions, ( English translation in St
 Petersburg Math. J
, 1992
"... numbers β> 0 is the function exp(−‖x ‖ β q) positive definite on R ⋉? Here q> 2 and ‖x‖q = (x1  q + · · ·+xn  q) 1/q. Denote by Bn(q) the set of such numbers β. ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
(Show Context)
numbers β> 0 is the function exp(−‖x ‖ β q) positive definite on R ⋉? Here q> 2 and ‖x‖q = (x1  q + · · ·+xn  q) 1/q. Denote by Bn(q) the set of such numbers β.
THE COMPLEX BUSEMANNPETTY PROBLEM ON SECTIONS OF CONVEX BODIES
"... Abstract. The complex BusemannPetty problem asks whether origin symmetric convex bodies in C n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4. 1. ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
Abstract. The complex BusemannPetty problem asks whether origin symmetric convex bodies in C n with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n ≤ 3 and negative if n ≥ 4. 1.
Distance geometry in quasihypermetric spaces
 I, Bull. Aust. Math. Soc
"... Abstract. Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X. Define ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X. Define
A twodimensional HahnBanach theorem
 Proc. Amer. Math. Soc
"... Abstract. Let ~T = ..."
(Show Context)
Sufficient enlargements in the study of projections in normed linear spaces
 Supplement), Proceedings, Dr. George Bachman Memorial Conference, The Allahabad Mathematical Society
"... Abstract. The study of sufficient enlargements of unit balls of Banach spaces forms a natural line of attack of some wellknown open problems of Banach space theory. The purpose of the paper is to present known results on sufficient enlargements and to state some open problems. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The study of sufficient enlargements of unit balls of Banach spaces forms a natural line of attack of some wellknown open problems of Banach space theory. The purpose of the paper is to present known results on sufficient enlargements and to state some open problems.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.