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On Bodies Associated with a Given Convex Body

Published online by Cambridge University Press:  20 November 2018

Endre Makai Jr.  and
Horst Martini
Endre Makai Jr.
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Pf 127, H-l364 Budapest, HUNGARY, makai@math-inst.hu
Horst Martini
Affiliation:
Fakultät für Mathematik Technische Universität Chemnitz-Zwickau, D-09107 Chemnitz, GERMANY, martini@mathematik.tu-chemnitz.de
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Abstract

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Let 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)).

Keywords

52A2052B11
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 4 , 01 December 1996 , pp. 448 - 459
Copyright
Copyright © Canadian Mathematical Society 1996

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