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A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast to previous works, our (rather simple) proofs are based on purely order theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.
We show that for any compact convex set $K$ in $mathbb{R}^d$ and any finite family $mathcal{F}$ of convex sets in $mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $mathcal{F}$ contains an isometric copy of $K$ of volume $1$
In this paper, we will show methods to interpret some rigid origami with higher degree vertices as the limit case of structures with degree-4 supplementary angle vertices. The interpretation is based on separating each crease into two parallel crease
Dynamical effects in general relativity have been finally, relatively recently observed by LIGOcite{2016LRR....19....1A}. To be able to measure these signals, great care has to be taken to minimize all sources of noise in the detector. One of the sou
Recent advances in helioseismology, numerical simulations and mean-field theory of solar differential rotation have shown that the meridional circulation pattern may consist of two or more cells in each hemisphere of the convection zone. According to
We propose a scheme to determine the energy-band dispersion of quasicrystals which does not require any periodic approximation and which directly provides the correct structure of the extended Brillouin zones. In the gap labelling viewpoint, this all