intersection body
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2019 ◽  
pp. 929-935
Author(s):  
Lijuan Liu
Keyword(s):  


2018 ◽  
Vol 12 (4) ◽  
pp. 416-436 ◽  
Author(s):  
Holly R. Cashman
Keyword(s):  


2015 ◽  
Vol 67 (1) ◽  
pp. 3-27
Author(s):  
M. Angeles Alfonseca ◽  
Jaegil Kim

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.



2014 ◽  
Vol 04 (10) ◽  
pp. 529-534
Author(s):  
Lingzhi Zhao ◽  
Jun Yuan


2011 ◽  
Vol 53 (3) ◽  
pp. 523-534 ◽  
Author(s):  
MATHIEU MEYER ◽  
SHLOMO REISNER

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).



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


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


2004 ◽  
Vol 47 (2) ◽  
pp. 246-256 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini

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.



1996 ◽  
Vol 39 (4) ◽  
pp. 448-459 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini

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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