We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.

Abstract Delone sets are discrete point sets X in $${\mathbb {R}}^d$$ R d characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest “empty ball” that can be inserted into the interstices of X. The regularity radius $${\hat{\rho }}_d$$ ρ ^ d is defined as the smallest positive number $$\rho $$ ρ such that each Delone set with congruent clusters of radius $$\rho $$ ρ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that $${\hat{\rho }}_{d}={\textrm{O}(d^2\log _2 d)}R$$ ρ ^ d = O ( d 2 log 2 d ) R as $$d\rightarrow \infty $$ d → ∞ , independent of r. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2r and those with full-dimensional sets of d-reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that $${\hat{\rho }}_{d}={\textrm{O}(d\log _2 d)}R$$ ρ ^ d = O ( d log 2 d ) R as $$d\rightarrow \infty $$ d → ∞ , independent of r.

Abstract For a locally finite set in $${{{\mathbb {R}}}}^2$$ R 2 , the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in $${{{\mathbb {R}}}}^2$$ R 2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970)).

Abstract. For a locally finite set, [Formula: see text], the [Formula: see text] th Brillouin zone of [Formula: see text] is the region of points [Formula: see text] for which [Formula: see text] is the [Formula: see text]th smallest among the Euclidean distances between [Formula: see text] and the points in [Formula: see text]. If [Formula: see text] is a lattice, the [Formula: see text]th Brillouin zones of the points in [Formula: see text] are translates of each other, and together they tile space. Depending on the value of [Formula: see text], they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in [Formula: see text], and the convergence of the maximum volume of a chamber to zero for the integer lattice.

In a recent paper, Garber, Gavrilyuk, and Magazinov [ Discrete Comput. Geom., 53 (2015), pp. 245--260] proposed a sufficient combinatorial condition for a parallelohedron to be affinely Voronoi. We show that this condition holds for all 5-dimensional Voronoi parallelohedra. Consequently, the Voronoi conjecture in $\mathbb{R}^5$ holds if and only if every 5-dimensional parallelohedron is combinatorially Voronoi. Here, by saying that a parallelohedron $P$ is combinatorially Voronoi, we mean that $P$ is combinatorially equivalent to a Dirichlet--Voronoi polytope for some lattice $\Lambda$, and this combinatorial equivalence is naturally translated into equivalence of the tiling by copies of $P$ with the Voronoi tiling of $\Lambda$. We also propose a new condition which, if satisfied by a parallelohedron $P$, is sufficient to infer that $P$ is affinely Voronoi. The condition is based on the new notion of the Venkov complex associated with a parallelohedron and cohomologies of this complex.