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1.
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 3 and negative if n ≥ 4. Since the answer is negative in most dimensions, it is natural to ask what conditions on the (n − 1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions. The result is the complex analogue of [16].   相似文献   

2.
The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in Rn 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 analysis 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. We discuss possible generalizations.  相似文献   

3.
The Busemann-Petty problem asks whether origin-symmetric convex bodies in Rn with smaller central hyperplane sections necessarily have smaller n-dimensional volume. It is known that the answer to this problem is affirmative if n?4 and negative if n?5. We study this problem in hyperbolic and spherical spaces.  相似文献   

4.
On the generalized Busemann-Petty problem   总被引:1,自引:0,他引:1  
The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ n with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem. This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)  相似文献   

5.
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in Cn with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if n?3 and negative if n?4.  相似文献   

6.
The Busemann-Petty problem asks whether symmetric convex bodies in n with smaller (n–1)-dimensional volume of central hyperplane sections necessarily have smaller n-dimensional volume. The answer to this problem is affirmative for n4 and negative for n5. In this paper we generalize the Busemann-Petty problem to essentially arbitrary measure in place of the volume. We also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n.Mathematics Subject Classification (2000): 52A15, 52A21, 52A38  相似文献   

7.
The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions and n, respectively, so that i + n. The answer essentially depends on . The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.  相似文献   

8.
The Busemann-Petty problem asks whether symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller n-dimensional volume. The solution has recently been completed, and the answer is affirmative if n?4 and negative if n?5. In this article we present a short proof of the affirmative result and its generalization using the Funk-Hecke formula for spherical harmonics.  相似文献   

9.
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝn with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve the lower dimensional Busemann-Petty problem in the hyperbolic space ℍn.  相似文献   

10.
The Busemann–Petty problem asks whether origin-symmetric convex bodies in Rn with smaller areas of all central hyperplane sections necessarily have smaller n-dimensional volume. The solution was completed in the end of the 1990s, and the answer is affirmative if n4 and negative if n5. Since the answer is negative in most dimensions, it is natural to ask what information about the volumes of central sections of two bodies does allow to compare the n-dimensional volumes of these bodies in all dimensions. In this article we give an answer to this question in terms of certain powers of the Laplace operator applied to the section function of the body.  相似文献   

11.
The complex Busemann-Petty problem asks whether origin symmetric convex bodies in with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if and negative if . In this article we show that the answer remains the same if the volume is replaced by an “almost” arbitrary measure. This result is the complex analogue of Zvavitch’s generalization to arbitrary measures of the original real Busemann-Petty problem. Received: 6 May 2008  相似文献   

12.
Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ n the oracle returns a “Yes” ifzS; whereas ifzS then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.  相似文献   

13.
A positive integern is called a pseudoprime ifn|2 n ?2 andn is composite. W. Sierpinski put forward the following problem: Do there infinitely many arithmetical progressions formed of four pseudoprimes? In this paper it is proved that answer to this problem is in the affirmative.  相似文献   

14.
The existence of homeomorphisms establishign an isometry of normalized Haar measures on (metrizable) compact groups is studied. In the case of 0-dimensional groups, a complete answer is given in terms of the indices of open normal subgroups. For example, for the countable powers of the groups ℤ/(m) and ℤ/(n), the answer is affirmative if and only ifm andn have the same prime divisors. A certain class of extensions of 0-dimensional groups is also studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 188–194, August, 2000.  相似文献   

15.
We express the volume of central hyperplane sections of star bodies inR n in terms of the Fourier transform of a power of the radial function, and apply this result to confirm the conjecture of Meyer and Pajor on the minimal volume of central sections of the unit balls of the spaces p n with 0<p<2. Research supported in part by the NSF Grant DMS-9531594.  相似文献   

16.
On the maximal ergodic theorem for certain subsets of the integers   总被引:6,自引:0,他引:6  
It is shown that the set of squares {n 2|n=1, 2,…} or, more generally, sets {n t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL 2-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL p-functions,p>1.  相似文献   

17.
Let N ≥ n + 1, and denote by K the convex hull of N independent standard gaussian random vectors in ℝn. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes. Supported by the Clay Mathematics Institute and by NSF grant #DMS-0456590  相似文献   

18.
It is known that non-symmetric convex bodies generally cannot be characterized by the volumes of hyperplane sections through one interior point. Falconer and Gardner, however, independently proved that volumes of hyperplane sections through two different interior points determine the body uniquely. We prove that if −1 < q < n − 1 is not an integer, then the derivatives of the order q at zero of parallel section functions at one interior point completely characterize convex bodies in . If 0 ≤ q < n − 1 is an integer then one needs the derivatives of order q at two different interior points (except for the case where q = n − 2, q odd), generalizing the results of Falconer and Gardner. The first named author was partially supported by the NSF grant DMS 0455696. Received: 31 January 2006  相似文献   

19.
An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n+1)-space is Euclidean complete for n≥2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R 3 must be an elliptic paraboloid. Oblatum 16-VI-2001 & 27-II-2002?Published online: 29 April 2002  相似文献   

20.
Univalent mappings associated with the Roper-Suffridge extension operator   总被引:15,自引:0,他引:15  
The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingFH(Bn). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iffS *, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB n. Finally, we investigate some open problems. Partially supported by the Natural Sciences and Engineering Research Council of Canada under grant A9221.  相似文献   

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