Dual Brunn-Minkowski Inequality Research Articles

Dual Brunn-Minkowski inequality for $ C $-star bodies

<abstract><p>In this paper, we introduced the concept of $ C $-star bodies in a fixed pointed closed convex cone $ C $ and studied the dual mixed volume for $ C $-star bodies. For $ C $-star bodies, we established the corresponding dual Brunn-Minkowski inequality, dual Minkowski inequality, and dual Aleksandrov-Fenchel inequality. Moveover, we found that the dual Brunn-Minkowski inequality for $ C $-star bodies can strengthen the Brunn-Minkowski inequality for $ C $-coconvex sets.</p></abstract>

The Dual Brunn–Minkowski Inequalities in Spherical and Hyperbolic Spaces

The Dual Brunn–Minkowski Inequalities in Spherical and Hyperbolic Spaces

Plurisubharmonic geodesics and interpolating sets

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of $${\mathbb {C}}^n$$ . Namely, two non-pluripolar, polynomially closed, compact subsets of $${\mathbb {C}}^n$$ are interpolated as level sets $$L_t=\{z: u_t(z)=-1\}$$ for the geodesic $$u_t$$ between their relative extremal functions with respect to any ambient bounded domain. The sets $$L_t$$ are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge–Ampere capacities of $$L_t$$ satisfy a dual Brunn–Minkowski inequality.

Orlicz dual affine quermassintegrals

Abstract In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the L p {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

The dual difference Aleksandrov–Fenchel inequality

Recently, a dual Minkowski inequality and a dual Brunn–Minkowski inequality for volume differences were established. Following this, in this paper we establish a dual Aleksandrov–Fenchel inequality for dual mixed volume differences which generalizes several recent results.

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.

Dual Quermassintegral Differences for Intersection Body

In this paper, we introduce the concept of dual quermassintegral differences. Further, we give the dual Brunn-Minkowski inequality and dual Minkowski inequality for dual quermassintegral differences for mixed intersection bodies.

A strong law of large numbers on the harmonic p-combination

In this paper, we establish a strong law of large numbers for the harmonic p-combinations of random star bodies. Starting from this theorem, we prove a strong law of large numbers in Lp space and provide the probabilistic version of dual Brunn-Minkowski inequality.

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.

Gaussian Brunn-Minkowski inequalities

A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coecients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.

The dual Brunn-Minkowski inequalities for star dual of mixed intersection bodies

In this article, some dual Brunn-Minkowski inequalities are established for star dual of mixed intersection bodies with respect to the harmonic p-combination and p-radial linear combination.

Dual Brunn–Minkowski inequality for volume differences

In this paper, we introduce the concepts of dual quermassintegral differences and width-integral differences, and discuss the theory of dual Brunn–Minkowski type for them. One of the results implies that for two star bodies which are dilations of each other, the dual Brunn–Minkowski inequality still holds after two arbitrary star bodies included in them being excluded, respectively.

Formula omitted]-dual affine surface area

According to the notion of L p -affine surface area by Lutwak, in this paper, we introduce the concept of L p -dual affine surface area. Further, we establish the affine isoperimetric inequality and the Blaschke–Santaló inequality for L p -dual affine surface area. Besides, the dual Brunn–Minkowski inequality for L p -dual affine surface area is presented.

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.

THE DUAL BRUNN-MINKOWSKI INEQUALITIES FOR INTERSECTION BODIES AND TWO ADDITIONS

In this paper, some dual Brunn-Minkowski inequalities are established for intersection bodies for the harmonic Blaschke additions and p-radial additions.

Inequalities for dual affine quermassintegrals

For star bodies, the dual affine quermassintegrals were introduced and studied in several papers. The aim of this paper is to study them further. In this paper, some inequalities for dual affine quermassintegrals are established, such as the Minkowski inequality, the dual Brunn-Minkowski inequality, and the Blaschke-Santaló inequality.

INEQUALITIES FOR MIXED INTERSECTION BODIES

In this paper, some properties of mixed intersection bodies are given, and inequalities from the dual Brunn-Minkowski theory (such as the dual Minkowski inequality, the dual Aleksandrov-Fenchel inequalities and the dual Brunn-Minkowski inequalities) are established for mixed intersection bodies.

Stability of Inequalities in the Dual Brunn-Minkowski Theory

Stability versions are given of several inequalities from E. Lutwak's dual Brunn-Minkowski theory. These include the dual Aleksandrov-Fenchel inequality, the dual Brunn-Minkowski inequality, and the dual isoperimetric inequality. Two methods are used. One involves the application of strong forms of Clarkson's inequality forLpnorms that hold for nonnegative functions, and the other utilizes a refinement of the arithmetic-geometric mean inequality. A new and more informative proof of the equivalence of the dual Brunn-Minkowski inequality and the dual Minkowski inequality is given. The main results are shown to be the best possible up to constant factors.