Some inequalities for Lp radial Blaschke-Minkowski homomorphisms1

The notion of radial Blaschke-Minkowski homomorphisms was presented by Schuster. Afterwards, Wang et al. introduced L_p radial Blaschke-Minkowski homomorphisms. In this paper, associated with L_p-dual a?ne surface areas, we establish some inequalities including the Brunn-Minkowski type inequality, cyclic inequality and monotonic inequalities, and give an a?rmative answer and a negative answer of Busemann-Petty problem for the L_p radial Blaschke-Minkowski homomorphisms.     

On radial Blaschke-Minkowski homomorphisms

In this article we first establish a dual isoperimetric type inequality for mixed radial Blaschke-Minkowski homomorphisms of different orders. Second, we prove a dual Brunn-Minkowski-type and a dual Minkowski-type inequality for mixed radial Blaschke-Minkowski homomorphisms.     

Lp径向Blaschke-Minkowski同态的Lp对偶几何表面积

2006年，Schuster提出了径向Blaschke-Minkowski同态的概念.随后，汪卫等人将其推广到L_p径向Blaschke-Minkowski同态.本文结合L_p对偶几何表面积，建立了L_p径向Blaschke-Minkowski同态的若干不等式，包括Brunn-Minkowski型不等式和单调不等式.并给出了L_p径向Blaschke-Minkowski同态的Busemann-Petty问题的肯定和否定形式.     

On Blaschke-Minkowski homomorphisms

In this article we establish a Brunn-Minkowski-type inequality for mixed Blaschke-Minkowski homomorphisms.     

The (p,q)-mixed geominimal surface areas1

Abstract

The (p, q)-mixed geominimal surface areas are introduced. A special case of the new concept is the L_p geominimal surface area introduced by Lutwak. Related inequalities, such as a?ne isoperimetric inequality, monotonous inequality, cyclic inequality, and Brunn-Minkowski inequality, are established. These new inequalities strengthen some well-known inequalities related to the L_p geominimal surface area.     

The Orlicz Brunn–Minkowski inequality

The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the _Lp$L_p$ Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the _Lp$L_p$ Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn–Minkowski inequality.     

Some inequalities for radial Blaschke-Minkowski homomorphisms

We establish some Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms with respect to Orlicz radial sums and differences of dual quermassintegrals.     

Lp-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.     

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

Capacities, surface area, and radial sums

A dual capacitary Brunn-Minkowski inequality is established for the (n−1)-capacity of radial sums of star bodies in Rn. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in Rn, 1?p<n, proved by Borell, Colesanti, and Salani. When n?3, the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition. Note that when n=3, the (n−1)-capacity is the classical electrostatic capacity. A proof is also given of both the inequality and a (different) equality condition when n=2. The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense (n−1)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

Reverse inequalities for zonoids and their application

We prove inequalities for mixed volumes of zonoids with isotropic generating measures. A special case is an inequality for zonoids that is reverse to the generalized Urysohn inequality, between mean width and another intrinsic volume; here the equality case characterizes parallelepipeds. We apply this to a question from stochastic geometry. It is known that among the stationary Poisson hyperplane processes of given positive intensity in n-dimensional Euclidean space, the ones with rotation invariant distribution are characterized by the fact that they yield, for k∈{2,…,n}, the maximal intensity of the intersection process of order k. We show that, if the kth intersection density is measured in an affine-invariant way, the processes of hyperplanes with only n fixed directions are characterized by a corresponding minimum property.     

You can recognize the shape of a figure from its shadows!

Let C ₁ and C ₂ be convex closed domains in the plane with C ² boundaries C ₁ and C ₂ intersecting each other in nonzero angles. Assume the two strictly convex bodies F ₁ and F ₂ with C ² boundaries in the interior of C ₁C ₂ subtend equal visual angles at each point of C ₁ and C ₂. Then F ₁ and F ₂ coincide. Generalizations are also discussed.Supported by the Hungarian NSF, OTKA Nr. T4427, W015425 and F016226.     

Inequalities for <Emphasis Type="Bold">cd</Emphasis>-Indices of Joins and Products of Polytopes

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For polytopes U and V, we determine explicit recurrences for computing the cd-index of the free join and the cd-index of the Cartesian product U x V. As an application of these recurrences, we prove the inequality involving the cd-indices of three polytopes.     

Extensions of an inequality of Bonnesen toD-dimensional space and curvature conditions for convex bodies

For convex bodies inE ^d (d 3) with diameter 2 we consider inequalitiesW _i – W _d–1 +( - 1) W _d 0 (i = 0, , d – 2) whereW _j are the quermassintegrals. In addition, for a ball, equality is attained for a body of revolution for which the elementary symmetric functions _d–1–i of main curvature radii is constant. The inequality is actually proved fori = d – 2 by means of Weierstrass's fundamental theorem of the calculus of variations.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Constructing a polytope to approximate a convex body

We develop an algorithm to construct a convex polytopeP withn vertices, contained in an arbitrary convex bodyK inR ^d , so that the ratio of the volumes |K/P|/|K| is dominated byc ·. d/n ^2/(d–1).Supported in part by the fund for the promotion of research in the Technion     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

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.