scholarly journals Dual Quermassintegral Differences for Intersection Body

2014 ◽  
Vol 04 (10) ◽  
pp. 529-534
Author(s):  
Lingzhi Zhao ◽  
Jun Yuan
Keyword(s):  
Intersection Body ◽  
Dual Quermassintegral

On the L p intersection body

2007 ◽  
Vol 28 (12) ◽  
pp. 1669-1678 ◽  
Author(s):  
Xian-yang Zhu ◽  
Gang-song Leng
Keyword(s):  
Intersection Body

scholarly journals On the Local Convexity of Intersection Bodies of Revolution

2015 ◽  
Vol 67 (1) ◽  
pp. 3-27
Author(s):  
M. Angeles Alfonseca ◽  
Jaegil Kim
Keyword(s):  
Convex Body ◽  
Convex Geometry ◽  
Euclidean Ball ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Body Of Revolution ◽  
Local Convexity ◽  
Bodies Of Revolution ◽  
Intersection Body ◽  
Intersection Bodies

AbstractOne of the fundamental results in convex geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach–Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

L p -dual Quermassintegral sums

2007 ◽  
Vol 50 (9) ◽  
pp. 1347-1360 ◽  
Author(s):  
Chang-jian Zhao
Keyword(s):  
Dual Quermassintegral

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)).

scholarly journals The unit ball is an attractor of the intersection body operator

2011 ◽  
Vol 226 (3) ◽  
pp. 2629-2642 ◽  
Author(s):  
Alexander Fish ◽  
Fedor Nazarov ◽  
Dmitry Ryabogin ◽  
Artem Zvavitch
Keyword(s):  
Unit Ball ◽  
Intersection Body

On Maximal k-Sections and Related Common Transversals of Convex Bodies

2004 ◽  
Vol 47 (2) ◽  
pp. 246-256 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Cross Section ◽  
Convex Bodies ◽  
Intersection Body ◽  
The Cross

AbstractGeneralizing results from [MM1] referring to the intersection body IK and the cross-section body CK of a convex body K ⊂ , d ≥ 2, we prove theorems about maximal k-sections of convex bodies, k ∈ {1, … ,d − 1}, and, simultaneously, statements about common maximal (d − 1)- and 1-transversals of families of convex bodies.

scholarly journals THE CONVEX INTERSECTION BODY OF A CONVEX BODY

2011 ◽  
Vol 53 (3) ◽  
pp. 523-534 ◽  
Author(s):  
MATHIEU MEYER ◽  
SHLOMO REISNER
Keyword(s):  
Convex Body ◽  
Interior Point ◽  
Radial Function ◽  
Hyperplane Section ◽  
Symmetric Body ◽  
Intersection Body ◽  
Centrally Symmetric

AbstractLet L be a convex body in n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI(L, z). This generalizes the classical intersection bodyI(L, z) (whose radial function at u ∈ Sn−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).

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