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.

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 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.

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.

Successive Minima and Radii

2009 ◽  
Vol 52 (3) ◽  
pp. 380-387 ◽  
Author(s):  
Martin Henk ◽  
María A. Hernández Cifre
Keyword(s):  
Convex Body ◽  
The Body ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Successive Minima

AbstractIn this note we present inequalities relating the successive minima of an o-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

scholarly journals Retrieving convex bodies from restricted covariogram functions

2007 ◽  
Vol 39 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Gennadiy Averkov ◽  
Gabriele Bianchi
Keyword(s):  
Convex Body ◽  
Lebesgue Measure ◽  
Dimensional Space ◽  
Convex Bodies ◽  
Baire Category ◽  
The Body ◽  
Planar Case ◽  
Large Classes ◽  
Negative Results ◽  
Planar Convex

The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

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) $.

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