Dual Mixed Volumes Research Articles

Characterizing the dual mixed volume via additive functionals

Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the dual mixed volume, the fundamental concept in the dual Brunn-Minkowski theory. The characterizations are shown to be best possible in the sense that none of the assumptions can be omitted. The results obtained are in the spirit of a similar characterization of the mixed volume in the classical Brunn- Minkowski theory, obtained recently by Milman and Schneider, but the methods employed are completely different.

L p Harmonic radial combinations of star bodies

For star bodies, the \(L_{p}\) harmonic radial combinations were defined and studied in several papers. In this paper, we study the mean chord of \(L_{p}\) harmonic radial combinations of star bodies and get an upper bound for dual mixed volumes of \(L_{p}\) harmonic radial combination of star bodies and their polar bodies. Furthermore, we obtain a dual Urysohn type inequality and a dual Bieberbach type inequality.

Characterization of dual mixed volumes via polymeasures

We prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated with it. To do this, we use tools from the theory of multilinear operators on spaces of continuous functions. Along the way we reprove, with these same techniques, a recently found characterization of the dual mixed volume.

Orlicz Dual Mixed Volumes

In the paper, we introduce Orlicz dual mixed volumes, and inequalities of Brunn–Minkowski type are obtained for Orlicz dual mixed volumes. The new Orlicz dual Brunn–Minkowski inequality implies the L p -dual Brunn–Minkowski inequality. A new Orlicz dual Minkowski inequality is obtained that generalizes the L p -dual Minkowski inequality.

TWO COMPLEX COMBINATIONS AND COMPLEX INTERSECTION BODIES

This paper devotes to establish complex dual Brunn-Minkowski theory. At first, we introduce the concepts of complex radial combination and complex radial-Blaschke combination, and obtain the relations between those two combinations and dual mixed volumes. Then, we extend the properties of real intersection body to the complex case. Finally, we prove some complex geometric inequalities about complex intersection bodies and complex mixed intersection bodies, such as dual Brunn-Minkowski type, dual Aleksandrov-Fenchel type and dual Minkowski type inequality. Moreover, as applications, we get some corollaries including an isoperimetric type inequality and a uniqueness theorem.

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in R m by introducing a renormalization of the r α − m -potential for α ⩽ 0 . This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.

On Minkowski's inequality and its application

In the paper, we first give an improvement of Minkowski integral inequality. As an application, we get new Brunn-Minkowski-type inequalities for dual mixed volumes. 2000 Mathematics Subject Classification: 52A30, 52A40, 26D15

L p -mixed intersection bodies and star duality

The paper extends the two notions of the dual mixed volumes and Lp-intersection body to q-dual mixed volumes and Lp-mixed intersection body, respectively. Inequalities for the star dual of Lp-mixed intersection bodies are established.

Inequalities of the quermassintegrals for the L p-projection body and the L p-centroid body

Associated with the concepts of the L p -mixed quermassintegrals, the L p -mixed volume, and the L p -dual mixed volume, we establish inequalities for the quermassintegrals of the L p -projection body and the L p -centroid body. Further, the general results for the Shephard problem of the L p -projection body and the L p -centroid body are obtained.

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.

Cross $i$-sections of star bodies and dual mixed volumes

In this paper, we establish an extension of Funk's section theorem. Our result has the following corollary: If K is a star body in R whose central i-slices have the same volume (with appropriate dimension) as the central i-slices of a centered body M, then the dual quermassintegrals satisfy Wj (M) ≤ Wj(K), for any 0 < j < n - i, with equality if and only if K = M. The case that K is a centered body implies Funk's section theorem.

Some dual kinematic formulas *

In this article, some kinematic formulas for dual quermassintegral of star bodies and for chord power integrals of convex bodies are established by using dual mixed volumes. These formulas are the extensions of the fundamental kinematic formula involving quermassintegral to the case of dual quermassintegral and chord power integrals.

From John to Gauss–John positions via dual mixed volumes

We investigate the optimization of the dual mixed volumes { W ˜ i ( S K ) , 0 ∈ S K ⊆ D n } where K ⊆ R n is a convex body, D n the Euclidean ball and SK runs over all positions of K. When S is linear we give necessary and sufficient conditions for K to be in extremal position in terms of a decomposition of the identity. We consider affine problems and we also present an approach involving parallel sections of K which can be understood as a dual fractional Kubota formula.

Dual mixed volumes and the slicing problem

We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and quotients of L p and related spaces. An extension of these results to negative values of p is also obtained, using generalized intersection-bodies. In particular, we show that the isotropic constant of a convex body which is contained in an intersection-body is bounded (up to a constant) by the ratio between the latter's mean-radius and the former's volume-radius. We also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure.

SOME INEQUALITIES ABOUT DUAL MIXED VOLUMES OF STAR BODIES

The authors establish some inequalities about the dual mixed volumes of star bodies in Rn. These inequalities are the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals.

Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn–Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey–Brunn–Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ℓ-position or isotropic position.

Harmonic radial combinations and dual mixed volumes

Harmonic radial combinations and dual mixed volumes

Star Valuations and Dual Mixed Volumes

Since its creation by Brunn and Minkowski, what has become known as the Brunn Minkowski theory has provided powerful machinery to solve a broad variety of inverse problems with stereological data. The machinery of the Brunn Minkowski theory includes mixed volumes (of Minkowski), symmetrization techniques (such as those of Steiner and Blaschke), isoperimetric inequalities (such as the Brunn Minkowski, Minkowski, and Aleksandrov Fenchel inequalities), integral transforms (such as the cosine transform), and important auxiliary bodies associated with these transforms (such as Minkowski's projection bodies). Schneider's recent book [22] on the Brunn Minkowski theory is the best available introduction to the subject. While the Brunn Minkowski theory has proven to be of enormous value in answering inverse questions regarding projections of convex bodies onto subspaces, the theory has been of little value in answering inverse questions with data regarding intersections with subspaces. However, recent advancements have been made in the development of a dual Brunn Minkowski theory [3, 4, 5, 6, 8, 9, 11, 13, 16, 23, 26, 27, 28, 29] which has been tailored specifically for dealing with such questions. In contrast to the Brunn Minkowski theory, in the dual theory convex bodies are replaced by star-shaped sets, and projections onto subspaces are replaced by intersections with subspaces. The machinery of the dual theory includes dual mixed volumes (introduced by Lutwak [15, 16]), dual isoperimetric inequalities ([16]), and important auxiliary bodies known as intersection bodies (first introduced by Busemann for special centered convex bodies in [1]; later defined for star-shaped sets by Lutwak [16]. See also [9]). A comprehensive introduction to geometric tomography, including the dual Brunn Minkowski theory, may be found in Gardner's book [17]. Unfortunately, there still remain fundamental and foundational problems with the dual Brunn Minkowski theory. One of the most beautiful and article no. 0048

Centered Bodies and Dual Mixed Volumes

We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in ${{\mathbf {R}}^n}$ by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central $(n - 1)$-slices. It implies Lutwak’s affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry.

Tomography of Convex and Star Bodies

New definitions of a star body and its radial function are given. These are used in studying sections of convex and star bodies, in conjunction with the dual mixed volumes of Lutwak, the spherical Radon transform, and the i-chord functions. (When the body contains the origin in its interior, and i ≠ 0, the latter equals the sum of the values of the ith power of the radial function at antipodal points in the unit sphere.) Special cases of results obtained include the following. A simple construction is given of non-spherical convex bodies of constant i-section, i.e., such that each i-dimensional section through an interior point has the same i-dimensional volume. It is shown that two star bodies, containing the origin in their interiors and with analytic boundaries, must be equal up to reflection in the origin, if the volumes of their sections by i-dimensional subspaces for two different values of i are equal. Examples are given to show that the last statement is no longer necessarily true if the boundaries are only infinitely differentiable or if the bodies are convex polytopes. It is also proved that a star body which has constant i-section for two different values of i, with respect to the same interior point, must be a ball centered at that point. This answers the dual of the old question as to whether there exist non-spherical convex bodies of constant width and constant brightness. New results are also obtained concerning the determination of convex polytopes by the volumes of their sections through one or more points.