Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

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.

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Centrally symmetric convex bodies and sections having maximal quermassintegrals

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.2.1197 ◽

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.

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A Blichfeldt-type inequality for centrally symmetric convex bodies

Monatshefte für Mathematik ◽

10.1007/s00605-012-0461-2 ◽

2012 ◽

Vol 170 (3-4) ◽

pp. 371-379 ◽

Cited By ~ 3

Author(s):

Matthias Henze

Keyword(s):

Convex Bodies ◽

Type Inequality ◽

Symmetric Convex ◽

Centrally Symmetric

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Almost ellipsoidal sections and projections of convex bodies

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051355 ◽

1975 ◽

Vol 77 (3) ◽

pp. 529-546 ◽

Cited By ~ 5

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.

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

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.2.1235 ◽

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.

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

Journal of Mathematics Research ◽

10.5539/jmr.v9n2p84 ◽

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.

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On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-044-8 ◽

2009 ◽

Vol 52 (3) ◽

pp. 407-415 ◽

Cited By ~ 4

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.

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