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.     

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.     

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.     

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].      

Comparison of volumes by means of the areas of central sections

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.     

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 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 complex Busemann-Petty problem for arbitrary measures

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     

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.     

The Busemann-Petty problem on entropy of log-concave functions

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....     

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 orthostochastic, unistochastic and qustochastic matrices

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size n?4.     

Hyperplane sections of convex bodies

We prove sharp inequalities for the volumes of hyperplane sections bisecting a convex body in Rn. This leads to a relative isoperimetric inequality for arbitrary hyperplane sections of a convex body.     

Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X. As an application, we reduce the study of invariant integrally closed ideals of V×X to that of some smaller variety in the case of X=M_n,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237-262] for two special infinite series.     

On approximation by projections of polytopes with few facets

We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a section of an N-dimensional simplex one has $$d(B,K) \geqslant c\sqrt {\frac{n}{{\ln \frac{{2N\ln (2N)}}{n}}}} $$ , where d(·, ·) denotes the Banach-Mazur distance and c is an absolute positive constant. The result is sharp up to a logarithmic factor.     

Macroscopic dimension and essential manifolds

M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.     

Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

We construct six unitary trace invariants for 2×2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We have discovered a curious trace identity for two unit-speed one-parameter subgroups of Sp(1). A modification gives an infinite family of trace identities for quaternions as well as for 2×2 complex matrices. We were not able to locate these identities in the literature. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n×n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.     

Projections of convex bodies and the fourier transform

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl _p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl _p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.