Blaschke Addition Research Articles

The Lp-mixed quermassintegrals for 0 < p < 1

Abstract In the paper, Lp-harmonic addition, p-harmonic Blaschke addition and Lp-dual mixed volume are improved. A new p-harmonic Blaschke mixed quermassintegral is introduced. The relationship between p-harmonic Blaschke mixed volume and Lp-dual mixed volume is shown.

Cyclic Brunn–Minkowski inequalities for dual mixed volumes

Combined with radial Minkowski addition, radial Blaschke addition and harmonic Blaschke addition, we establish cyclic Brunn–Minkowski inequalities for dual mixed volumes of star bodies.

Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications

The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.

The Dual Blaschke Addition

It is remarkable that there is a duality in geometric tomography between results on projections of convex bodies and results on sections of star (rather than convex) bodies. The radial Blaschke addition, which is the dual version of Blaschke addition, as an operation between central symmetric star bodies is introduced in this paper. The relationship between it and the classical radial addition, many properties of radial Blaschke addition and related inequalities are established.

Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.

New inequalities for star bodies

In this paper, we investigate the radial addition and Blaschke addition and get some new Brunn-Minkowski inequalities associated with dual quermassintegrals and chord integral for star bodies.

Dual cyclic Brunn-Minkowski inequalities

New cyclic Brunn-Minkowski inequalities for dual quermassintegrals of star bodies are established. These inequalities are obtained with respect to radial Minkowski addition as well as radial Blaschke and harmonic Blaschke addition.

The Brunn-Minkowski type inequalities for mixed brightness-integrals

The mixed brightness-integrals were defined by Li and Zhu. In this paper, we first establish two Brunn-Minkowski inequalities of the mixed brightness-integrals based on the Blaschke sum and Minkowski sum of convex bodies. Further, we also obtain the Beckenbach-Dresher type inequalities of the mixed brightness-integrals combining the harmonic Blaschke sum and the harmonic radial sum of star bodies.

A characterization of Blaschke addition

A characterization of Blaschke addition as a map between origin-symmetric convex bodies is established. This results from a new characterization of Minkowski addition as a map between origin-symmetric zonoids, combined with the use of Lévy–Prokhorov metrics. A full set of examples is provided that show the results are in a sense the best possible.

On quotients of $i$th affine surface areas

Following the volume dierence function, we rst introduce the notion of the ane surface area quotient function. We establish Brunn{Minkowski type inequalities for the ane surface area quotient function, which in special function, ane surface area quotient function, Blaschke sum, Brunn{Minkowski inequal- ity

Valuations on Sobolev spaces

All affinely covariant convex-body-valued valuations on the Sobolev space $W^{1,1}({\Bbb{R}}^n)$ are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function $f\in W^{1,1}({\Bbb{R}}^n)$ the unit ball of its optimal Sobolev norm.

Minimal pairs of polytopes and their number of vertices

We define what is called Blaschke difference for polytopes as an inverse operation to Blaschke addition. Using this operation we give a new algorithm to reduce and find a minimal pair of polytopes from the given class of the Radstrom-Hormander lattice containing a pair of polytopes in IR 2 . This method gives a better algorithmic insight and easy to handle than the one given by Handschug (1989). We also prove that a pair of polytopes in the plane is minimal if and only if the sum of the number of their vertices is minimal in the class. However, it is shown in the paper that, this last statement does not hold true in general for higher dimensional spaces.

Brunn-Minkowski inequalities for star duals of intersection bodies and two additions

In this paper, we first establish the dual Brunn-Minkowski inequality for the star duals for the Lp radial sum. Furthermore, we give some Brunn-Minkowski inequalities for the star duals of intersection bodies for the Lp radial sum and the Lp harmonic Blaschke sum.

Uniformly computable aspects of inner functions: estimation and factorization

AbstractThe theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type‐Two Effectivity as our foundation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Uniformly Computable Aspects of Inner Functions

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also very useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We give a uniformly computable version of Frostman's Theorem. We then claim that the Factorization Theorem is not uniformly computably true. We then claim that for an inner function u, the Blaschke sum of u provides the exact amount of information necessary to compute the factorization of u. Along the way, we discuss some uniform computability results for Blaschke products. These results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation.

Convolutions and multiplier transformations of convex bodies

Rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator

Convex set symmetry measurement using Blaschke addition

In this paper we state a new way for measuring symmetry of convex sets. The measures introduced are based on the Blaschke sum. This operation has some important properties, and is a natural way for combining convex objects when represented by their extended gaussian image. Different kinds of symmetry are considered and their properties outlined. To form the Blaschke sum, it is necessary to solve the Minkowski reconstruction problem. We study the method of reconstruction and give some improvements to make the scheme more efficient.