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1.
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
2.
Boris Rubin 《Advances in Mathematics》2010,225(3):1461-1498
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.
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
4.
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
5.
Alexander Koldobsky 《Advances in Applied Mathematics》2004,33(4):728-732
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
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.
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
8.
The i th section function of a star body in
n
gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin,
by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the
origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast
with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak.
Received December 4, 1996, and in revised form April 14, 1997. 相似文献
9.
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
10.
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
11.
The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝ
n
, inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator.
Research of this author was partially supported by KBN Grant no. 1 P03A 015 27.
This author holds the Canada Research Chair in Geometric Analysis. 相似文献
12.
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
13.
For a convex body K ⊂ ℝn and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K
0 ⊂ ℝn be smooth convex bodies with boundaries of class C
2 and positive Gauss-Kronecker curvature and assume K
0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K
0 are homothetic if their i-th and j-th projection functions are proportional. When K
0 is a Euclidean ball this shows that a convex body with C
2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball.
The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953. 相似文献
14.
This paper deals with a reducible sℓ(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture:
Suppose that sℓ(2,C) acts on the formal power series ring via (1.1). Then I(f) = (ℓ
i1) ⊕ (ℓ
i2) ⊕... ⊕ (ℓ
is
) modulo some one dimensional sℓ(2,C) representations where (ℓ
i
) is an irreducible sℓ(2,C) representation of ℓ
i
dimension and {ℓ
i1
ℓ
i2,...,ℓ
is
} ⊆ {ℓ
1
, ℓ
2...,ℓ
r
}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat
the reducible action and higher dimensional representations successively. 相似文献
15.
Matthias Kriesell 《Graphs and Combinatorics》2006,22(4):481-485
Let κ(G) denote the (vertex) connectivity of a graph G. For ℓ≥0, a noncomplete graph of finite connectivity is called ℓ-critical if κ(G−X)=κ(G)−|X| for every X⊆V(G) with |X|≤ℓ.
Mader proved that every 3-critical graph has diameter at most 4 and asked for 3-critical graphs having diameter exceeding
2. Here we give an affirmative answer by constructing an ℓ-critical graph of diameter 3 for every ℓ≥3. 相似文献
16.
E. A. Gorin 《Mathematical Notes》2000,68(2):167-172
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
17.
We present a formula for the Fourier transforms of order statistics in ℝ
n
showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in ℝ
n
.
Fora
1≥...≥a
n≥0 andq>0, denote by ℓ
w,q
n
then-dimensional Lorentz space with the norm ‖(x
1,...,x
n)‖=(a
1(x
1
*
)
q
+...+a
n(x
n
*
)
q
)1/q
, where (x
1
*
,...,x
n
*
) is the non-increasing permutation of the numbers |x
1|,...,|x
n|. We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spaces intoL
q [10] to prove that, forn≥3 andq≤1, the space ℓ
w,q
n
is isometric to a subspace ofL
q if and only if the numbersa
1,...,a
n form an arithmetic progression. Forq>1, all the numbersa
i must be equal so that ℓ
w,q
n
= ℓ
q
n
. Consequently, the Lorentz function spaceL
w,q(0, 1) is isometric to a subspace ofL
q if and only ifeither 0<q<∞ and the weightw is a constant function (so thatL
w,q=Lq),or q≤1 andw(t) is a decreasing linear function.
Finally, we relate our results to the theory of positive definite functions.
Both authors were supported in part by the NSF Workshop in Linear Analysis and Probability held at Texas A&M University in
August 1993. The work was done during the first author’s visit to Texas A&M University. 相似文献
18.
19.
Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful
quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist,
the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily
close to one. We will answer this question in the affirmative for ℓ
1-localized frames. We then specialize our results to Gabor multi-frames with generators in M
1(R
d
), and Gabor molecules with envelopes in W(C, l
1). As a main tool in this work, we show there is a universal function g(x) so that, for every ε =s> 0, every Parseval frame {f
i
}
i=1
M
for an N-dimensional Hilbert space H
N
has a subset of fewer than (1+ε)N elements which is a frame for H
N
with lower frame bound g(ε/(2M/N − 1)). This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition,
the results give compelling new evidence in support of a general definition of redundancy given in [5]. 相似文献
20.
Maite Fernández-Unzueta 《Israel Journal of Mathematics》2012,188(1):301-322
We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived
from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach
spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product
of the entire spaces. We show that the ℓ
p
spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2. 相似文献