8. Difference, Body, and Race

2020 ◽  
pp. 131-147
Keyword(s):  
Difference Body

Rogers and Shephard inequality for the Orlicz difference body

2015 ◽  
Vol 143 (9) ◽  
pp. 4029-4039 ◽  
Author(s):  
Fangwei Chen ◽  
Wenxue Xu ◽  
Congli Yang
Keyword(s):  
Difference Body

The difference body of a convex body

1957 ◽  
Vol 8 (3) ◽  
pp. 220-233 ◽  
Author(s):  
C. A. Rogers ◽  
G. C. Shephard
Keyword(s):  
Convex Body ◽  
Difference Body ◽  
The Difference

scholarly journals How do difference bodies in complex vector spaces look like? A geometrical approach

2015 ◽  
Vol 17 (04) ◽  
pp. 1450023 ◽  
Author(s):  
Judit Abardia ◽  
Eugenia Saorín Gómez
Keyword(s):  
Complex Structure ◽  
Classical Case ◽  
The Body ◽  
Geometrical Approach ◽  
Complex Vector ◽  
Geometrical Properties ◽  
Complex Difference ◽  
Difference Body ◽  
The Difference ◽  
Two Bodies

We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a ball, we prove that it is a polytope if and only if the two bodies involved in the construction are polytopes and provide several inequalities for classical magnitudes of the complex difference body, as volume, quermassintegrals and diameter, in terms of the corresponding ones for the involved bodies.

CHORD POWER INTEGRALS OF SIMPLICES

2009 ◽  
Vol 02 (04) ◽  
pp. 557-565
Author(s):  
Wing-Sum Cheung ◽  
Ge Xiong
Keyword(s):  
Type Inequality ◽  
Projection Body ◽  
Polar Projection ◽  
Difference Body ◽  
Dual Quermassintegrals

In this paper, we obtain a formula relating the chord power integrals of a simplex K and the dual quermassintegrals of its difference body DK. As interesting applications, we express the volumes of difference body DK and polar projection body Π*K in terms of the volume of simplex K. Santaló-type inequality for chord power integrals of simplex is also established.

scholarly journals Sections of the Difference Body

2000 ◽  
Vol 23 (1) ◽  
pp. 137-146 ◽  
Author(s):  
M. Rudelson
Keyword(s):  
Difference Body ◽  
The Difference

Difference, Body, and Race

2014 ◽  
pp. 131-147
Author(s):  
Michelle A. Gonzalez
Keyword(s):  
Difference Body

On Bodies Associated with a Given Convex Body

1996 ◽  
Vol 39 (4) ◽  
pp. 448-459 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Cross Section ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Shadow Boundary ◽  
Projection Body ◽  
Intersection Body ◽  
Open Set ◽  
Difference Body ◽  
The Difference

AbstractLet d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J. Gardner), while for d ≥ 3 the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric. If IK = c ˙ CK for some constant c > 0, then K is centred, and if both IK and CK are centred balls, then K is a centred ball. If the chordal symmetral and the difference body of K are constant multiples of each other, then K is centred; if both are centred balls, then K is a centred ball. For d ≥ 3 we determine the minimal number of facets, and estimate the minimal number of vertices, of a convex d-polytope P having no plane shadow boundary with respect to parallel illumination (this property is related to the inclusion [CP] ⊂ int(ΠP)).

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