共查询到20条相似文献,搜索用时 15 毫秒
1.
Marisa Zymonopoulou 《Archiv der Mathematik》2008,91(5):436-449
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
2.
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
3.
Alexander Koldobsky 《Advances in Mathematics》2003,177(1):105-114
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
4.
Science China Mathematics - The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ?nwith smaller central hyperplane sections necessarily have smaller volumes.... 相似文献
5.
V. Yaskin 《Advances in Mathematics》2006,203(2):537-553
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
6.
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
7.
On an analytic generalization of the Busemann-Petty problem 总被引:1,自引:0,他引:1
Songjun Lv 《Journal of Mathematical Analysis and Applications》2008,341(2):1438-1444
In this paper, we establish an extension of the connections between an analytic generalization of the Busemann-Petty problem and the positive definite distributions. Our results show that the structure of the positive definite distributions in Rn is closely related to the analytic generalization of the Busemann-Petty problem which was posed by Koldobsky. 相似文献
8.
The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ
n
with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes
of central hyperplane sections so that the answer becomes affirmative in all dimensions.
The first-named author was supported in part by the NSF grant DMS-0136022 and by a grant from the University of Missouri Research
Board. 相似文献
9.
10.
Boris Rubin 《Israel Journal of Mathematics》2009,173(1):213-233
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 ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. 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
11.
We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions. 相似文献
12.
Marisa Zymonopoulou 《Positivity》2009,13(4):717-733
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
13.
14.
The point in polygon problem for arbitrary polygons 总被引:11,自引:0,他引:11
A detailed discussion of the point in polygon problem for arbitrary polygons is given. Two concepts for solving this problem are known in literature: the even–odd rule and the winding number, the former leading to ray-crossing, the latter to angle summation algorithms. First we show by mathematical means that both concepts are very closely related, thereby developing a first version of an algorithm for determining the winding number. Then we examine how to accelerate this algorithm and how to handle special cases. Furthermore we compare these algorithms with those found in literature and discuss the results. 相似文献
15.
A. Koldobsky 《Israel Journal of Mathematics》1999,110(1):75-91
It is proved that for arbitrarymεℕ and for a sufficiently nontrivial compact groupG of operators acting on a “typical”n-dimensional quotientX
n
ofl
1
m
withm=(1+δ)n, there is a constantc=c(δ) such that
Supported in part by KBN grant no. 2 P03A 034 10. 相似文献
16.
AbstractFirst, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers. 相似文献
17.
N. M. Neskorodev 《Journal of Mathematical Sciences》1998,92(5):4128-4132
By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce
the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional
problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent
variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external
forces.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 11–19. 相似文献
18.
V. Yaskin 《Journal of Geometric Analysis》2006,16(4):735-745
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
19.
We solve a variation of the cell-growth problem by enumerating (unlabeled) polygonal clusters, whose constituent cells are all n-gons. The corresponding labeled problem had already been solved by one of us and its solution provides an initial step in the procedure developed here. It will be seen that when n = 3, this amounts to counting triangulations of the disk. 相似文献
20.
Mu‐Tao Wang 《纯数学与应用数学通讯》2004,57(2):267-281
Let Ω be a bounded C2 domain in ?n and ? ?Ω → ?m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?m with f|?Ω = ? and with the graph of f a minimal submanifold in ?n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?m satisfies 8nδ supΩ |D2ψ| + √2 sup?Ω |Dψ| < 1, then the Dirichlet problem for ψ|?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc. 相似文献