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.

Contributions to a General Theory of View-Obstruction Problems

1993 ◽  
Vol 45 (3) ◽  
pp. 517-536 ◽  
Author(s):  
V. C. Dumir ◽  
R. J. Hans-Gill ◽  
J. B. Wilker
Keyword(s):  
Critical Parameter ◽  
Original Problem ◽  
Complete Solution ◽  
Convex Bodies ◽  
Positive Cone ◽  
Symmetric Convex ◽  
Reflection Symmetry ◽  
Centrally Symmetric ◽  
Minimal Blocking ◽  
Lower Dimensional

AbstractIn the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

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.

scholarly journals On the volume product of planar polar convex bodies — Lower estimates with stability

2013 ◽  
Vol 50 (2) ◽  
pp. 159-198
Author(s):  
K. Böröczky ◽  
E. Makai ◽  
M. Meyer ◽  
S. Reisner
Keyword(s):  
Convex Body ◽  
Rotational Symmetry ◽  
Polar Body ◽  
Convex Bodies ◽  
Stability Estimate ◽  
The Body ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
The Stability ◽  
Area Product

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(K − K)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

2018 ◽  
Vol 148 (5) ◽  
pp. 913-937
Author(s):  
Dorin Bucur ◽  
Ilaria Fragalà
Keyword(s):  
Convex Body ◽  
Polar Body ◽  
Convex Bodies ◽  
Type Inequality ◽  
Symmetric Convex ◽  
Cheeger Constant ◽  
Regular Triangle ◽  
Centrally Symmetric ◽  
Affine Image ◽  
Krahn Inequality

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

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 On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies

2009 ◽  
Vol 52 (3) ◽  
pp. 407-415 ◽  
Author(s):  
Zsolt Lángi ◽  
Márton Naszódi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric ◽  
The One ◽  
Homothetic Copies

AbstractThe Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ℝd is 2d. Naszódi proved that the quantity in question is not larger than 2d+1. We present an improvement to this result by proving the upper bound 3 · 2d–1 for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

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

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
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