Intersection Bodies Research Articles

Intersection bodies of polytopes: translations and convexity

We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of I(P+t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I(P+t)$$\\end{document} can be extended to polynomials in variables t∈Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in \\mathbb {R}^d$$\\end{document} within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions.

On the polar dualities and star dualities of the quasi Lp -intersection bodies

Abstract Yu, Wu and Leng defined the quasi L p {L_{p}} -intersection bodies. In this paper, we consider the polar dualities and star dualities of quasi L p {L_{p}} -intersection bodies and establish some related inequalities.

The Transfigurations/Twists of the Religious: An Introduction to the Theological Epistemology of Marcella Althaus-Reid

Latin American cuir/questioning theology is a recent field of critical studies that not only draws on queer/questioning thought and activism but also contributes to contemporary sex-gendered dissident proposals. This article presents some epistemic contributions from the transfigurations, twists, “indecencies,” and resignifications of the religious. It is presented as an example of this theological integration to the program of the Doctorate in Critical Gender Studies of the Universidad Iberoamericana. In particular, the proposal of this article aims to analyze the intersection body–gender–religion from a hermeneutic approach of the book Dios cuir, by the Argentinean theologian Marcella Althaus-Reid.

Comparison problems for Radon transforms

Given two non-negative functions f and g such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it follow that the Lp-norm of f is smaller than the Lp-norm of g for a given p>1? We consider this problem for the classical and spherical Radon transforms. In both cases we point out classes of functions for which the answer is affirmative, and show that in general the answer is negative if the functions do not belong to these classes. The results are in the spirit of the solution of the Busemann-Petty problem from convex geometry, and the classes of functions that we introduce generalize the class of intersection bodies introduced by Lutwak in 1988. We also deduce slicing inequalities that are related to the well-known Oberlin-Stein type estimates for the Radon transform.

Convexity of the radial sum of a star body and a ball

AbstractWe investigate the convexity of the radial sum of two convex bodies containing the origin. Generally, the radial sum of two convex bodies containing the origin is not convex. We show that the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a pair of convex bodies containing the origin whose radial sum is convex.We also investigate the convexity of the intersection body of a convex body containing the origin. Generally, the intersection body of a convex body containing the origin is not convex. Busemann’s theorem states that the intersection body of any centered convex body is convex. We are interested in how to construct convex intersection bodies from convex bodies without any symmetry (especially, central symmetry). We show that the intersection body of the radial sum of a star body (with respect to the origin) and any large centered ball is convex, which produces a convex body with no symmetries whose intersection body is convex.

Stability related to the Lp$L_p$ Busemann–Petty problem

AbstractThe Busemann–Petty problem for asks: does imply for origin‐symmetric convex bodies K and L? Here, is the volume of K and the operator is the intersection body of K. In this paper, we study the linear stability and separation results related to the Busemann–Petty problem. A typical result we obtained is: for small enough, if K is an intersection body and L is a star body such that for any , the following inequality: implies where is the Minkowski functional of , and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Busemann–Petty problem.

Intersection bodies of polytopes

We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for its computation. Moreover, we compute the irreducible components of the algebraic boundary and provide an upper bound for the degree of these components.

Inequalities on the (p,q)-mixed volume involving Lp centroid bodies and Lp intersection bodies

In this paper, applying for the Minkowski?s and H?lder?s integral inequalities, we obtain four theorems about the (p, q)-mixed volume involving the Lp centroid bodies and the Lp intersection bodies, respectively. The former two theorems reveal the convexity of the functionals related to the (p, q)-mixed volume, in terms of the dual Blaschke addition introduced in [Journal of Geometric Analysis, 30 (2020) 3026-3034], and the latter two theorems expose the monotonicity of the other functionals related to the (p, q)-mixed volume.

The dual Orlicz Brunn–Minkowski inequality for the intersection body

The dual Orlicz Brunn–Minkowski inequality for the intersection body

SL(n) covariant function-valued valuations

Classifications of SL(n) covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, Lp moment bodies, Lp difference bodies, and polar Lp moment bodies (L−p intersection bodies). Using the new classifications, we also establish some Euler-type relations.

Lutwak–Petty projection inequalities for Minkowski valuations and their duals

Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for intersection bodies of all orders by Leng and Lu is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.

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.

Busemann-Petty problems for L_p mixed intersection bodies

Busemann-Petty problems for L_p mixed intersection bodies

Estimating volume and surface area of a convex body via its projections or sections

The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least outer volume ratio distance from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

The logarithmic intersection body

The logarithmic intersection body

Generalized Intersection Bodies with Parameter

AIn 2005, the classical intersection bodies and Lp intersection bodies were extended. Afterwards, the concept of general Lp intersection bodies and the generalized intersection bodies was introduced. In this paper, we define the generalized bodies with parameter. Besides, we establish the extremal values for volume, Brunn-Minkowski type inequality for radial combination and Lp harmonic Blaschke combination of this notion.

EODL: Energy Optimized Distributed Localization Method in three-dimensional underwater acoustic sensors networks

The localization represents one of the most important aspects of underwater acoustic sensor networks, offering three-dimensional monitoring of marine environment, military detection and attack missions. This paper proposes the Energy Optimized Distributed Localization Method (EODL) as an efficient technique for the distributed sensor network localization based on the mobile beacon water. This technique employs an Autonomous Underwater Vehicle (AUV) as the mobile beacon. The moving path of the AUV in the three-dimensional water environment and the time interval of the broadcast transmitting are found by applying a qualitative analysis of the superiority of the rectangular path. In such a technique, the unknown sensor node position is calculated by monitoring beacon signals without requiring time synchronization between nodes. We consider the innermost intersection body for the localization process, to calculate the geometric intersection relation of different information signal ranges. Then, we calculate the location information of the nodes. Moreover, the localization error analysis of the method will be presented. We compare the simulation results of the proposed method (EODL) with the LDSN. In addition, the experimental results verify that the proposed method introduces less error and higher coverage rate comparing with the conventional methods.

On mixed complex intersection bodies

On mixed complex intersection bodies

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

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 Lp-Brunn-Minkowski inequality for the radial mixed Blaschke-Minkowski homomorphisms and the mixed intersection bodies, respectively.

The normalized L_p intersection bodies

The normalized L_p intersection bodies