Intersection Bodies Research Articles

MIXED CHORD-INTEGRALS OF STAR BODIES

The mixed chord-integrals are defined. The Fenchel-Aleksandrov inequality and a general isoperimetric inequality for the mixed chordintegrals are established. Furthermore, the dual general Bieberbach inequality is presented. As an application of the dual form, a Brunn-Minkowski type inequality for mixed intersection bodies is given.

$p$-Minkowski inequalities for mixed intersection bodies

$p$-Minkowski inequalities for mixed intersection bodies

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.

Convexity of [formula omitted]-intersection bodies

We extend the classical Brunn theorem to symmetric moments of convex bodies and use it to prove the convexity of the L p -intersection bodies of centered convex bodies.

L p-Brunn-Minkowski inequality

We introduce the notion of L p -mixed intersection body ( p < 1) and extend the classical notion dual mixed volume to an L p setting. Further, we establish the Brunn-Minkowski inequality for the q-dual mixed volumes of star duals of L p -mixed intersection bodies.

On intersection and mixed intersection bodies

Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies.

Valuations and Busemann–Petty type problems

Projection and intersection bodies define continuous and GL ( n ) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann–Petty problem. We consider the question whether Φ K ⊆ Φ L implies V ( K ) ⩽ V ( L ) , where Φ is a homogeneous, continuous operator on convex or star bodies which is an SO ( n ) equivariant valuation. Important previous results for projection and intersection bodies are extended to a large class of valuations.

Intersection bodies and generalized cosine transforms

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.

Generalized intersection bodies are not equivalent

In [A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000) 1507–1526], A. Koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent. The structures of these two types of generalized intersection bodies have been studied by the author in [E. Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2) (2006) 530–567], providing substantial evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in L p for certain negative values of p.

Formula omitted] intersection bodies

Basic relations and analogies between intersection bodies and their symmetric and nonsymmetric L p counterparts are established.

On the L p intersection body

On the L p intersection body

On the local equatorial characterization of zonoids and intersection bodies

In this paper we show that there is no local equatorial characterization of zonoids in odd dimensions. This gives a negative answer to the conjecture posed by W. Weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions. In addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies.

Quasi L p -Intersection Bodies

The purpose of this paper is to generalize the notion of intersection bodies to that of quasi L p -intersection bodies. The L p -analogs of the Busemann intersection inequality and the Brunn–Minkowski inequality for the quasi L p -intersection bodies are obtained. The Aleksandrov–Fenchel inequality for the mixed quasi L p -intersection bodies is also established.

On dual quermassintegrals of mixed intersection bodies

In this paper, it is shown that a family of inequalities involving mixed intersection bodies holds. The Busemann intersection inequality is the first of this family. All of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body.

Formula omitted] intersection bodies

In this paper, we generalize the dual Brunn–Minkowski inequality and monotonicity of intersection bodies to L p intersection bodies.

INEQUALITIES FOR STAR DUALS OF INTERSECTION BODIES

In this paper, we present a new kind of duality between intersection bodies and projection bodies. Furthermore, we establish some counterparts of dual Brunn-Minkowski inequalities for intersection bodies.

A note on 𝑘-intersection bodies

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has also played an important role in the solution of the Busemann-Petty problem. A more general concept of k k -intersection bodies is related to the generalization of the Busemann-Petty problem. In this note, we compare classes of k k -intersection bodies for different k k and examine the conjecture that these classes increase with k k . In particular, we construct a 4 4 -intersection body that is not a 2 2 -intersection body.

On Star Duality of Mixed Intersection Bodies

A new kind of duality between intersection bodies and projection bodies is presented. Furthermore, some inequalities for mixed intersection bodies are established. A geometric inequality is derived between the volumes of star duality of star bodies and their associated mixed intersection integral.

Volume Inequalities and Additive Maps of Convex Bodies

Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Minkowski inequality for the volume of polar projection bodies.

Intersection bodies and valuations

All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called "intersection operator"—an operator that played a critical role in the solution of the Busemann-Petty problem.