Large planar Poisson–Voronoi cells containing a given convex body

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.

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Bounds on the Lattice Point Enumerator via Slices and Projections

Discrete & Computational Geometry ◽

10.1007/s00454-021-00310-7 ◽

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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Tetrahedra of varying density and their applications

The Visual Computer ◽

10.1007/s00371-021-02189-0 ◽

2021 ◽

Author(s):

Dennis R. Bukenberger ◽

Hendrik P. A. Lensch

Keyword(s):

Tetrahedral Mesh ◽

Rotational Inertia ◽

Exact Mass ◽

Accurate Analysis ◽

Exact Integration ◽

Voronoi Cells ◽

Current State ◽

Chebyshev Norm ◽

Exact Calculations ◽

Vector Matrix

Abstract We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve $$L_{2|\infty }$$ L 2 | ∞ Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations. Graphic abstract

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Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

Advances in Applied Probability ◽

10.1017/s0001867800009733 ◽

2000 ◽

Vol 32 (01) ◽

pp. 1-18 ◽

Cited By ~ 5

Author(s):

F. Baccelli ◽

K. Tchoumatchenko ◽

S. Zuyev

Keyword(s):

Communication Networks ◽

Mobile Networks ◽

Path Length ◽

Euclidean Distance ◽

Voronoi Tessellation ◽

Voronoi Cells ◽

Ergodic Properties ◽

Potential Applications ◽

The Mean ◽

Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

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An extremal property of the hypersphere

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100026542 ◽

1951 ◽

Vol 47 (1) ◽

pp. 245-247 ◽

Cited By ~ 33

Author(s):

A. M. Macbeath

Keyword(s):

Convex Body ◽

Extremal Property ◽

Plane Case ◽

Simple Application ◽

Plane Convex ◽

Plane Convex Body

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

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Measuring the Surface Area of a Convex Body

Annals of Mathematics ◽

10.2307/1969305 ◽

1944 ◽

Vol 45 (4) ◽

pp. 793 ◽

Cited By ~ 7

Author(s):

P. A. P. Moran

Keyword(s):

Convex Body ◽

Surface Area

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