Affine-regular hexagons of extreme areas inscribed in a centrally symmetric convex body

2003 ◽  
Vol 3 (1) ◽  
Author(s):  
Marek Lassak
Keyword(s):  
Convex Body ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Centrally Symmetric

scholarly journals Almost ellipsoidal sections and projections of convex bodies

1975 ◽  
Vol 77 (3) ◽  
pp. 529-546 ◽  
Author(s):  
D. G. Larman ◽  
P. Mani
Keyword(s):  
Convex Body ◽  
High Dimension ◽  
Interior Point ◽  
Convex Bodies ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Centrally Symmetric ◽  
Dimensional Section ◽  
Projections Of Convex Bodies

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

On the Inequality for Volume and Minkowskian Thickness

2006 ◽  
Vol 49 (2) ◽  
pp. 185-195 ◽  
Author(s):  
Gennadiy Averkov
Keyword(s):  
Convex Body ◽  
Geometric Inequality ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Centrally Symmetric ◽  
Arbitrary Convex ◽  
Simple Corollary ◽  
The Difference ◽  
The Right ◽  
Finite Dimensional Banach Space

AbstractGiven a centrally symmetric convex body B in , we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ball B. Let K be an arbitrary convex body in ℳd(B). The relationship between volume V(K) and the Minkowskian thickness (= minimal width) ΔB(K) of K can naturally be given by the sharp geometric inequality V(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain that with equality on the left attained if and only if B is the difference body of a simplex and on the right if B is a cross-polytope. The main result of this paper is that for d = 2 the equality on the right implies that B is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

scholarly journals Bounds for Covering Symmetric Convex Bodies by a Lattice Congruent to a Given Lattice

2017 ◽  
Vol 9 (2) ◽  
pp. 84
Author(s):  
Beomjong Kwak
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Lattice Covering ◽  
Centrally Symmetric

In this paper, we focus on lattice covering of centrally symmetric convex body on $\mathbb{R}^2$. While there is no constraint on the lattice in many other results about lattice covering, in this study, we only consider lattices congruent to a given lattice to retain more information on the lattice. To obtain some upper bounds on the infimum of the density of such covering, we will say a body is a coverable body with respect to a lattice if such lattice covering is possible, and try to suggest a function of a given lattice such that any centrally symmetric convex body whose area is not less than the function is a coverable body. As an application of this result, we will suggest a theorem which enables one to apply this to a coverable body to suggesting an efficient lattice covering for general sets, which may be non-convex and may have holes.

scholarly journals Visibility in crowds of translates of a centrally symmetric convex body

2010 ◽  
Vol 31 (3) ◽  
pp. 710-719
Author(s):  
Javier Alonso ◽  
Horst Martini ◽  
Margarita Spirova
Keyword(s):  
Convex Body ◽  
Symmetric Convex Body ◽  
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 Extremum properties of the new ellipsoid

2007 ◽  
Vol 38 (2) ◽  
pp. 159-165 ◽  
Author(s):  
Yuan Jun ◽  
Si Lin ◽  
Leng Gangsong
Keyword(s):  
Convex Body ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
New Ellipsoid

For a convex body $ K $ in $ {\mathbb R}^{n} $, Lutwak, Yang and Zhang defined a new ellipsoid $ \Gamma_{-2}K $, which is the dual analog of the Legendre ellipsoid. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $ K $, there exist an ellipsoid $ E $ and a parallelotope $ P $ such that $ \Gamma_{-2}E \supseteq \Gamma_{-2}K \supseteq \Gamma_{-2}P $ and $ V(E)=V(K)=V(P) $; (ii) For any convex body $K$ whose John point is at the origin, then there exists a simplex $T$ such that $ \Gamma_{-2}K \supseteq \Gamma_{-2}T $ and $ V(K)=V(T) $.

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.

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