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

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.

scholarly journals Covering a convex body by its negative homothetic copies

2001 ◽  
Vol 197 (1) ◽  
pp. 43-51
Author(s):  
Janusz Januszewski ◽  
Marek Lassak
Keyword(s):  
Convex Body ◽  
Homothetic Copies

Covering the boundary of a convex body with its smaller homothetic copies

2019 ◽  
Vol 342 (2) ◽  
pp. 393-404
Author(s):  
Dejing Lv ◽  
Senlin Wu ◽  
Liping Yuan
Keyword(s):  
Convex Body ◽  
Homothetic Copies

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.

Equation of state for hard convex body fluid mixtures

1998 ◽  
Vol 94 (5) ◽  
pp. 809-814 ◽  
Author(s):  
C. BARRIO ◽  
J.R. SOLANA
Keyword(s):  
Equation Of State ◽  
Convex Body ◽  
Body Fluid ◽  
Fluid Mixtures

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

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