The modified complex Busemann-Petty problem on sections of convex bodies

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].      

Modified Busemann-Petty problem on sections of convex bodies

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.     

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

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.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

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.     

The Busemann-Petty problem in hyperbolic and spherical spaces

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.     

On the generalized Busemann-Petty problem

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)     

The Busemann-Petty problem for arbitrary measures

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     

The complex Busemann-Petty problem on sections of convex bodies

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.     

The Busemann-Petty problem via spherical harmonics

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.     

A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space

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.     

Generalizations of the Busemann-Petty problem for sections of convex bodies

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.     

On the Determination of Star and Convex Bodies by Section Functions

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.     

Low dimensional sections versus projections of convex bodies

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.     

On bodies with directly congruent projections and sections

Let K and L be two convex bodies in R⁴, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfies an additional condition and some projections do not have certain π-symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.     

An application of the fourier transform to sections of star bodies

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 spacesℓ _p ⁿ with 0<p<2. Research supported in part by the NSF Grant DMS-9531594.     

The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces

Let V be a vector space of dimension n+1 over a field of p ^t elements. A d-dimensional subspace and an e-dimensional subspace are considered to be incident if their intersection is not the zero subspace. The rank of these incidence matrices, modulo p, are computed for all n, d, e and t. This result generalizes the well-known formula of Hamada for the incidence matrices between points and subspaces of given dimensions in a finite projective space. A generating function for these ranks as t varies, keeping n, d and e fixed, is also given. In the special case where the dimensions are complementary, i.e., d+e=n+ 1, our formula improves previous upper bounds on the size of partial m-systems (as defined by Shult and Thas).     

Towards the arithmetic degree of line bundles¶ on abelian varieties

In this paper we analyze the integral of the star-product of (n+1) Green currents associated to (n+1) global sections of an ample line bundle equipped with a translation invariant metric over an n-dimensional, polarized abelian variety. The integral is shown to equal the logarithm of the Petersson norm of a certain Siegel modular form, which is explicitly described in terms of the given data. This result can be interpreted as evaluating an archimedian height on a family of polarized abelian varieties. The key ingredient to the proof of the main formula is a dd ^c -variational formula for the integral under consideration. In the case of dimensions n=1,2,3 explicit examples in terms of classical Riemann theta functions are given. Received: 13 February 1998     

Sections of the Difference Body

Let K be an n -dimensional convex body. Define the difference body by K-K= { x-y | x,y ∈ K }. We estimate the volume of the section of K-K by a linear subspace F via the maximal volume of sections of K parallel to F . We prove that for any m -dimensional subspace F there exists x ∈ \bf R ⁿ , such that for some absolute constant C . We show that for small dimensions of F this estimate is exact up to a multiplicative constant. Received May 6, 1998, and in revised form July 23, 1998.     

An LMI description for the cone of Lorentz-positive maps II

Let L _n be the n-dimensional second-order cone. A linear map from ? ^m to ? ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n,?m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n???1)(m???1) that describes this cone.     

Gemischte volumina in Kanalscharen

A canal class of convex bodies inn-dimensional Euclidean space consists of all convex bodies which have the same orthogonal projection on some hyperplane. In such a canal class, improved versions of the general Brunn-Minkowski theorem and of the Aleksandrov-Fenchel inequalities for mixed volumes are valid. Partial results on the equality cases are obtained. As an application, a translation theorem of the Aleksandrov-Fenchel-Jessen type is proved.