scholarly journals Banach–Mazur Distance from the Parallelogram to the Affine-Regular Hexagon and Other Affine-Regular Even-Gons

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Marek Lassak
Keyword(s):  
Convex Bodies ◽  
Regular Hexagon ◽  
The Family ◽  
Centrally Symmetric ◽  
Planar Convex

AbstractWe show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is $$\frac{3}{2}$$ 3 2 and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just $$\frac{3}{2}$$ 3 2 . A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric bodies is at most $$\frac{3}{2}$$ 3 2 . Analogously, we deal with the Banach–Mazur distances between the parallelogram and the remaining affine-regular even-gons.

scholarly journals A measure of axial symmetry of centrally symmetric convex bodies

2010 ◽  
Vol 121 (2) ◽  
pp. 295-306 ◽  
Author(s):  
Marek Lassak ◽  
Monika Nowicka
Keyword(s):  
Axial Symmetry ◽  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

Transversals with Residue in Moderately Overlapping T(k)-Families of Translates

2009 ◽  
Vol 52 (3) ◽  
pp. 388-402
Author(s):  
Aladár Heppes
Keyword(s):  
Asymptotic Behavior ◽  
Compact Convex ◽  
Line Transversal ◽  
Type Result ◽  
Empty Interior ◽  
Disjoint Family ◽  
Straight Line ◽  
The Family ◽  
Centrally Symmetric ◽  
Line Meeting

AbstractLet K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. The integer k is the stabbing level of the family. Two translates Ki = K + ci and Kj = K + cj are said to be σ-disjoint if σK + ci and σK + cj are disjoint. A recent Helly-type result claims that for every σ > 0 there exists an integer k(σ) such that if a family of σ-disjoint unit diameter discs has property T(k)|k ≥ k(σ), then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval K. The asymptotic behavior of k(σ) for σ → 0 is considered as well.Katchalski and Lewis proved the existence of a constant r such that for every pairwise disjoint family of translates of an oval K with property T(3) a straight line can be found meeting all but at most r members of the family. In the second part of the paper σ-disjoint families of translates of K are considered and the relation of σ and the residue r is investigated. The asymptotic behavior of r(σ) for σ → 0 is also discussed.

A characterization of centrally symmetric convex bodies in En

1981 ◽  
Vol 10 (1-4) ◽  
pp. 161-176 ◽  
Author(s):  
D. G. Larman ◽  
N. K. Tamvakis
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

Some problems concerning centrally symmetric convex bodies

1978 ◽  
Vol 10 (3) ◽  
pp. 454-460
Author(s):  
V. A. Zalgaller ◽  
V. N. Sudakov
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

scholarly journals Centrally symmetric convex bodies and sections having maximal quermassintegrals

2012 ◽  
Vol 49 (2) ◽  
pp. 189-199
Author(s):  
E. Makai ◽  
H. Martini
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
The Body ◽  
Symmetric Convex ◽  
Local Theorem ◽  
Centrally Symmetric ◽  
Euclidean Unit Ball ◽  
Analogous Theorem ◽  
Lower Dimensional ◽  
Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

scholarly journals Self-packing of centrally symmetric convex bodies in ℝ2

1992 ◽  
Vol 8 (2) ◽  
pp. 171-189 ◽  
Author(s):  
P. G. Doyle ◽  
J. C. Lagarias ◽  
D. Randall
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric
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