scholarly journals Interaction of Poisson hyperplane processes and convex bodies

2019 ◽  
Vol 56 (4) ◽  
pp. 1020-1032 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Random Polytope ◽  
Mean Width ◽  
The Mean

AbstractGiven a stationary and isotropic Poisson hyperplane process and a convex body K in ${\mathbb R}^d$ , we consider the random polytope defined by the intersection of all closed half-spaces containing K that are bounded by hyperplanes of the process not intersecting K. We investigate how well the expected mean width of this random polytope approximates the mean width of K if the intensity of the hyperplane process tends to infinity.

scholarly journals The Characteristic Properties of the Minimal Lp-Mean Width

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Tongyi Ma
Keyword(s):  
Convex Body ◽  
Existence And Uniqueness ◽  
Convex Bodies ◽  
Smooth Convex ◽  
Smooth Convex Body ◽  
Minimal Position ◽  
Mean Width ◽  
Suitable Measure ◽  
Minimum Position

Giannopoulos proved that a smooth convex body K has minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal Lp-mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal Lp-mean width of convex bodies and prove the existence and uniqueness of the minimal Lp-mean width in its SL(n) images. In addition, we establish a characterization of the minimal Lp-mean width, conclude the average Mp(K) with a variation of the minimal Lp-mean width position, and give the condition for the minimum position of Mp(K).

scholarly journals The Mean Width of Circumscribed Random Polytopes

2010 ◽  
Vol 53 (4) ◽  
pp. 614-628 ◽  
Author(s):  
Károly J. Böröczky ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Asymptotic Formula ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Simplicial Polytope ◽  
Mean Width ◽  
The Mean ◽  
The Difference ◽  
Precise Asymptotic

AbstractFor a given convex body K in ℝd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic formula for the difference of the mean widths of P(n) and P is obtained.

scholarly journals Sections of Convex Bodies in John’s and Minimal Surface Area Position

2021 ◽  
Author(s):  
David Alonso-Gutiérrez ◽  
Silouanos Brazitikos
Keyword(s):  
Minimal Surface ◽  
Surface Area ◽  
Convex Bodies ◽  
Regular Simplex ◽  
Symmetric Convex ◽  
Polar Bodies ◽  
Mean Width ◽  
Centrally Symmetric ◽  
The Mean ◽  
Minimal Surface Area

Abstract We prove several estimates for the volume, the mean width, and the value of the Wills functional of sections of convex bodies in John’s position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John’s position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.

On characterizations of sausages via inequalities and roots of Steiner polynomials

2017 ◽  
Vol 17 (4) ◽  
Author(s):  
Jesús Yepes Nicolás
Keyword(s):  
Isoperimetric Inequality ◽  
Convex Bodies ◽  
Extremal Sets ◽  
Mean Width ◽  
The Family ◽  
Algebraic Properties ◽  
The Mean

AbstractWe prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.

On the volume of caps and bounding the mean-width of an isotropic convex body

2010 ◽  
Vol 149 (2) ◽  
pp. 317-331 ◽  
Author(s):  
PETER PIVOVAROV
Keyword(s):  
Convex Body ◽  
Lower Bounds ◽  
Tail Behavior ◽  
Linear Functionals ◽  
Converse Statement ◽  
Isotropic Position ◽  
Mean Width ◽  
The Mean

AbstractLetKbe a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting onKexhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if anarbitraryisotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

scholarly journals The mean width of random polytopes circumscribed around a convex body

2010 ◽  
Vol 81 (2) ◽  
pp. 499-523 ◽  
Author(s):  
Károly J. Böröczky ◽  
Ferenc Fodor ◽  
Daniel Hug
Keyword(s):  
Convex Body ◽  
Random Polytopes ◽  
Mean Width ◽  
The Mean

Excess entropy of the systems of molecules differing in size and shape

1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Equation Of State ◽  
Hard Spheres ◽  
Excess Entropy ◽  
Convex Bodies ◽  
Pure Substance ◽  
Liquid Equilibrium ◽  
Simple Rules ◽  
Different Shapes ◽  
The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

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