On Bodies Associated with a Given Convex Body
Canadian mathematical bulletin, Tome 39 (1996) no. 4, pp. 448-459
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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)).
Jr., Endre Makai; Martini, Horst. On Bodies Associated with a Given Convex Body. Canadian mathematical bulletin, Tome 39 (1996) no. 4, pp. 448-459. doi: 10.4153/CMB-1996-053-7
@article{10_4153_CMB_1996_053_7,
author = {Jr., Endre Makai and Martini, Horst},
title = {On {Bodies} {Associated} with a {Given} {Convex} {Body}},
journal = {Canadian mathematical bulletin},
pages = {448--459},
year = {1996},
volume = {39},
number = {4},
doi = {10.4153/CMB-1996-053-7},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-053-7/}
}
TY - JOUR AU - Jr., Endre Makai AU - Martini, Horst TI - On Bodies Associated with a Given Convex Body JO - Canadian mathematical bulletin PY - 1996 SP - 448 EP - 459 VL - 39 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-053-7/ DO - 10.4153/CMB-1996-053-7 ID - 10_4153_CMB_1996_053_7 ER -
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