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THE COMPLEX BUSEMANN-PETTY PROBLEM ON SECTIONS OF CONVEX BODIES
"... Abstract. The complex Busemann-Petty 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. ..."
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Abstract. The complex Busemann-Petty 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.
Adams inequalities in measure spaces
, 2009
"... In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite me ..."
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Cited by 8 (2 self)
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In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of R n. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve Adams’ results to functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including: sharp higher order Moser-Trudinger trace inequalities, sharp Adams/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for sums of weighted potentials, and for operators in the CR setting.
The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry
, 2006
"... The lower dimensional Busemann-Petty problem asks, whether n-dimensional origin-symmetric convex bodies, having smaller i-dimensional sections, necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. For i> 3, the answer is negative. For i = 2 and i = 3, the problem is st ..."
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The lower dimensional Busemann-Petty problem asks, whether n-dimensional origin-symmetric convex bodies, having smaller i-dimensional sections, necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. For i> 3, the answer is negative. For i = 2 and i = 3, the problem is still open, except when the body with smaller sections is a body of revolution. In this case the answer is affirmative. The paper contains a complete solution to the problem in the more general situation, when the body with smaller sections is invariant under orthogonal transformations preserving coordinate subspaces R ℓ and R n−ℓ of R n for arbitrary fixed 0 < ℓ < n.
COMPARISON OF VOLUMES OF CONVEX BODIES IN REAL, COMPLEX, AND QUATERNIONIC SPACES
, 2009
"... The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes of hy ..."
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The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic n-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic n-dimensional space has an affirmative answer if and only if n = 2. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed.
INTERSECTION BODIES AND GENERALIZED COSINE TRANSFORMS
, 2007
"... Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. We review some known facts and give them new proofs. The main focus is interrelation betwee ..."
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Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. We review some known facts and give them new proofs. The main focus is interrelation between generalized cosine transforms of different kinds and their application to investigation of certain family of intersection bodies, which we call λ-intersection bodies. The latter include k-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of Lp-spaces. In particular, we show that restriction of the spherical Radon transforms and the generalized cosine transforms onto lower dimensional subspaces preserves their integralgeometric structure. We apply this result to the study of sections of λ-intersection bodies. A number of new characterizations of this class of bodies and examples are given.
ON STRICT INCLUSIONS IN HIERARCHIES OF CONVEX BODIES.
, 707
"... Abstract. Let Ik be the class of convex k-intersection bodies in Rn (in the sense of Koldobsky) and Im k be the class of convex origin-symmetric bodies all of whose m-dimensional central sections are k-intersection bodies. We show that 1) Im k ̸ ⊂ Im+1 k, k + 3 ≤ m < n, and 2) Il ̸ ⊂ Ik, 1 ≤ k &l ..."
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Abstract. Let Ik be the class of convex k-intersection bodies in Rn (in the sense of Koldobsky) and Im k be the class of convex origin-symmetric bodies all of whose m-dimensional central sections are k-intersection bodies. We show that 1) Im k ̸ ⊂ Im+1 k, k + 3 ≤ m < n, and 2) Il ̸ ⊂ Ik, 1 ≤ k < l < n − 3.
COMPLEX INTERSECTION BODIES
"... Abstract. 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 a ..."
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Abstract. 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 Busemann-Petty problem for arbitrary measures and the corresponding hyperplane inequality for measures of complex intersection bodies. 1.
