Convolutions, Transforms, and Convex Bodies

The paper studies convex bodies and star bodies in Rⁿ by usingRadon transforms on Grassmann manifolds, p-cosine transformson the unit sphere, and convolutions on the rotation group ofRⁿ. It presents dual mixed volume characterizations of i-intersectionbodies and L_p-balls which are related to certain volume inequalitiesfor cross sections of convex bodies. It considers approximationsof special convex bodies by analytic bodies and various finitesums of ellipsoids which preserve special geometric properties.Convolution techniques are used to derive formulas for mixedvolumes, mixed surface measures, and p-cosine transforms. Theyare also used to prove characterizations of geometric functionals,such as surface area and dual quermassintegrals. 1991 MathematicsSubject Classification: 52A20, 52A40.     

Minimax detection of a signal for nondegenerate loss functions, and extremal convex problems

We study a wide class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for a special class of loss functions. Under rather general assumptions, we obtain the exact asymptotics (of Gaussian type) for the minimax error probabilities and the structure of asymptotically minimax tests. The methods are based on a reduction of the problems under consideration to extremal problems of minimization of a certain Hilbert norm on a convex set of sequences of probability measures on the real line. These extremal problems are investigated in a paper by I. A. Suslina for alternatives having the type of l_q-ellipsoids with l_p-balls removed. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 162–188.     

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.     

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.     

A functional analytic approach to intersection bodies

We consider several generalizations of the concept of an intersection body and show their connections with the Fourier transform and embeddings in L _p -spaces. These connections lead to generalizations of the recent solution to the Busemann—Petty problem on sections of convex bodies. Submitted: October 1999, Revision: December 1999.     

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.     

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 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.     

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].      

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 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....     

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.     

Fitting hyperplanes by minimizing orthogonal deviations

Hyperplanes withm + 1 parameters are fitted by minimizing the sum of weighted orthogonal deviations to a set ofN points. There is no inverse regression incompatibility. For unweighted orthogonall ₁-fits essentially the same number of points are on either side of an optimal hyperplane. The criterion function is neither convex, nor concave, nor even differentiable. The main result is that each orthogonall _p -fit interpolates at leastm + 1 points, for 0 <p 1. This enables the combinatorial strategy of systematically trying all possible hyperplanes which interpolatem + 1 data points.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

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.     

Extremal Polygons with Minimal Perimeter

In this paper we will investigate an isoperimetric type problem in lattices. If K is a bounded O-symmetric (centrally symmetric with respect to the origin) convex body in En of volume v(K) = 2n det L which does not contain non-zero lattice points in its interior, we say that K is extremal with respect to the given lattice L. There are two variations of the isoperimetric problem for this class of polyhedra. The first one is: Which bodies have minimal surface area in the class of extremal bodies for a fixed n-dimensional lattice? And the second one is: Which bodies have minimal surface area in the class of extremal bodies with volume 1 of dimension n? We characterize the solutions of these two problems in the plane. There is a consequence of these results, the solutions of the above problems in the plane give the solution of the lattice-like covering problem: Determine those centrally symmetric convex bodies whose translated copies (with respect to a fixed lattice L) cover the space and have minimal surface area.

     

A simple proof of an approximation theorem of H. Minkowski

It is proved that every convex bodyC inR ⁿ can be approximated by a sequenceC _k of convex bodies, whose boundary is the intersection of a level set of a homogeneous polynomial of degree 2k and a hyperplane. The Minkowski functional ofC _k is given explictly. Some further nice properties of the approximantsC _k are proved.Supported in part by BSF and Erwin Schrödinger Auslandsstipendium J0630.     

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.     

On two finite covering problems of Bambah,Rogers, Woods and Zassenhaus

Bambah, Rogers, Woods, and Zassenhaus considered the general problem of covering planar convex bodiesC byk translates of a centrally-symmetric convex bodyK ofE ² with the ramification that these translates cover the convex hullC _k of their centres. They proved interesting inequalities for the volume ofC andC _k . In the present paper some analogous results in euclideand-spaceE ^d are given. It turns out that on one hand extremal configurations ford5 are of quite different type than in the planar case. On the other hand inequalities similar to the planar ones seem to exist in general. Inequalities in both directions for the volume and other quermass-integrals are given.     

The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space \Bbb R^d{\Bbb R}^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices.