Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms

Abstract

In this paper, we extend the Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms to an Orlicz setting and an Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms is established. The new Orlicz-Brun-Minkowski inequality in special case yields the L_p-Brunn-Minkowski inequality for the radial mixed Blaschke-Minkowski homomorphisms and the mixed intersection bodies, respectively.     

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.     

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 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.     

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.     

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.     

L p Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms

In this article, some Brunn–Minkowski type inequalities for (radial) Blaschke–Minkowski homomorphisms with respect to (radial) L _p Minkowski addition are established.     

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.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

Orlicz–John ellipsoids

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved _Lp$L_p$ John ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.     

Sections and projections of homothetic convex bodies

For a pair of convex bodies K₁ and K₂ in Euclidean space , n ≥ 3, possibly unbounded, we show that K₁ is a translate of K₂ if either of the following conditions holds: (i) the orthogonal projections of K₁ on 2-dimensional planes are translates of the respective orthogonal projections of K₂, (ii) there are points p₁ ∈K₁ and p₂ ∈K₂ such that for every pair of parallel 2-dimensional planesL₁and L₂ through p₁ and p₂, respectively, the section K₁ ∩ L₁is a translate of K₂ ∩ L₂.     

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.     

Clarkson and Random Clarkson Inequalities for Lr(X)

We first show how (p,p′) Clarkson inequality for a Banach space X is inherited by Lebesgue-Bochner spaces L_r(X), which extends Clarkson's procedure deriving his inequalities for L_p from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in L_r(X) for any 1 ≤ r ≤ ∞; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for L_p are both directly derived from the parallelogram law for scalars.     

Mutual estimates of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norms and the Bellman function

In this paper, we describe the range of the L^p-norm of a function under fixed L^p-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two L^p-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse H?lder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5 Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 81–138.     

L p -Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

The B-spline representation for divided differences is used, for the first time, to provide L _p -bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities. The major result is the inequality where H_Θ f is the Hermite interpolant to f at the multiset of n points Θ, and is the diameter of . This inequality significantly improves upon Beesack's inequality, on which almost all the bounds given over the last 30 years have been based. Date received: June 24, 1994 Date revised: February 4, 1996.     

Bernstein–Nikolskii and Plancherel–Polya inequalities in Lp ‐norms on non‐compact symmetric spaces

By using Bernstein‐type inequality we define analogs of spaces of entire functions of exponential type in L_p (X), 1 ≤ p ≤ ∞, where X is a symmetric space of non‐compact. We give estimates of L_p ‐norms, 1 ≤ p ≤ ∞, of such functions (the Nikolskii‐type inequalities) and also prove the L_p ‐Plancherel–Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of “measurements” in a stable way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some discrete Sobolev's inequalities in three-dimensional spherical and cylindrical coordinates

The discrete Sobolev's inequalities inL _p norm are proved for three-dimensional spherical and cylindrical coordinates, by using discrete Hölder inequality, property of the triangle functions and complicated deduction.     

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.