scholarly journals On the relative lengths of sides of convex polygons

2003 ◽  
Vol 40 (1-2) ◽  
pp. 115-120
Author(s):  
Zs. Lángi
Keyword(s):  
Convex Body ◽  
Euclidean Distance ◽  
Upper Bounds ◽  
Relative Distance ◽  
Convex Polygons ◽  
Euclidean Length

Let C be a convex body. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. The aim of the paper is to find upper bounds for the minimum of the relative lengths of the sides of convex hexagons and heptagons.

scholarly journals On the relative distances of six points in a plane convex body

2005 ◽  
Vol 42 (3) ◽  
pp. 253-264
Author(s):  
Károly Böröczky ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Relative Distance ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

Determining convex polygons from their covariograms

2002 ◽  
Vol 34 (2) ◽  
pp. 261-266 ◽  
Author(s):  
Gabriele Bianchi
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Partial Answer ◽  
Convex Polygons ◽  
Geometric Covariogram

Knowing the (geometric) covariogram of a convex body is equivalent to knowing, for each direction u, the distribution of the lengths of the chords of that body which are parallel to u. We prove that the covariogram determines convex polygons, among all convex bodies, up to translation and reflection. This gives a partial answer to a problem posed by Matheron.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (3) ◽  
pp. 605-619 ◽  
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

Let K be a d-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by Kn the convex hull of n points chosen randomly and independently from K according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of Kn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

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 Further inequalities for the (generalized) Wills functional

2020 ◽  
pp. 2050011
Author(s):  
David Alonso-Gutiérrez ◽  
María A. Hernández Cifre ◽  
Jesús Yepes Nicolás
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
General Setting ◽  
Concave Function ◽  
Upper Bounds ◽  
Edge Length ◽  
Symmetric Convex ◽  
Concave Functions ◽  
Intrinsic Volumes

The Wills functional [Formula: see text] of a convex body [Formula: see text], defined as the sum of its intrinsic volumes [Formula: see text], turns out to have many interesting applications and properties. In this paper, we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for [Formula: see text] in terms of the volume of [Formula: see text], as well as Brunn–Minkowski and Rogers–Shephard-type inequalities for this functional. We also show that the cube of edge-length 2 maximizes [Formula: see text] among all [Formula: see text]-symmetric convex bodies in John position, and we reprove the well-known McMullen’s inequality [Formula: see text] using a different approach.

scholarly journals Intrinsic volumes of inscribed random polytopes in smooth convex bodies

2010 ◽  
Vol 42 (03) ◽  
pp. 605-619
Author(s):  
I. Bárány ◽  
F. Fodor ◽  
V. Vígh
Keyword(s):  
Convex Body ◽  
Convex Hull ◽  
Convex Bodies ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Lower And Upper Bounds ◽  
Covering Theorem ◽  
Intrinsic Volumes ◽  
Large Numbers ◽  
Distribution Matching

LetKbe ad-dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote byKnthe convex hull ofnpoints chosen randomly and independently fromKaccording to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of thesth intrinsic volumesVs(Kn) ofKnfors∈ {1,…,d}. Furthermore, strong laws of large numbers are proved for the intrinsic volumes ofKn. The essential tools are the economic cap covering theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.

On covering radius of codes over R = ℤ2 + uℤ2, where u2 = 0 using chinese euclidean distance

2017 ◽  
Vol 09 (02) ◽  
pp. 1750017 ◽  
Author(s):  
P. Chella Pandian
Keyword(s):  
Euclidean Distance ◽  
Upper Bounds ◽  
Covering Radius ◽  
Lower And Upper Bounds ◽  
Text Type ◽  
Simplex Codes ◽  
Type Code

In this paper, we give lower and upper bounds on the covering radius of codes over the ring [Formula: see text] where [Formula: see text] with respect to Chinese Euclidean distance and also obtain the covering radius of various Repetition codes, Simplex codes of [Formula: see text]-Type code and [Formula: see text]-Type code. We give bounds on the covering radius for MacDonald codes of both types over [Formula: see text]

scholarly journals The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

2021 ◽  
Author(s):  
Giulia Codenotti ◽  
Francisco Santos ◽  
Matthias Schymura
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Arbitrary Dimension ◽  
Upper Bounds ◽  
Discrete Analog ◽  
Covering Radius ◽  
Lattice Polytopes ◽  
Direct Sums ◽  
Discrete Surface ◽  
Covering Minimum

AbstractWe explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

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