Translative covering a convex body by its homothetic copies

2003 ◽  
Vol 40 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Janusz Januszewski
Keyword(s):  
Convex Body ◽  
Planar Convex ◽  
Translative Covering ◽  
Homothetic Copies

Any sequence of positive homothetic copies of a planar convex body C with total area not smaller than 6.5 times the area of C permits a translative covering of C.

Covering a convex body by smaller homothetic copies

1993 ◽  
Vol 45 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Valeriu Soltan ◽  
Éva Vásárhelyi
Keyword(s):  
Convex Body ◽  
Homothetic Copies

scholarly journals Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter

2014 ◽  
Vol 418 (2) ◽  
pp. 1030-1046 ◽  
Author(s):  
Antonio Cañete ◽  
Cinzia Miori ◽  
Salvador Segura Gomis
Keyword(s):  
Convex Body ◽  
Rotationally Symmetric ◽  
Planar Convex

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 Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Antonio Cañete
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Rotationally Symmetric ◽  
Planar Convex ◽  
Cheeger Sets

AbstractIn this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

On Planar Continuous Families of Curves

1969 ◽  
Vol 21 ◽  
pp. 513-530 ◽  
Author(s):  
Tudor Zamfirescu
Keyword(s):  
Convex Body ◽  
General Setting ◽  
The Other ◽  
Continuous Family ◽  
Families Of Curves ◽  
Family Of Curves ◽  
Planar Convex

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

Translative covering by homothetic copies

1993 ◽  
Vol 46 (2) ◽  
pp. 173-180 ◽  
Author(s):  
V. Bálint ◽  
A. Bálintová ◽  
M. Branická ◽  
P. Grešák ◽  
I. Hrinko ◽  
...  
Keyword(s):  
Translative Covering ◽  
Covering By Homothetic Copies ◽  
Homothetic Copies
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