The Kepler Conjecture 2025, 43 - Definition and Orientation of Voronoi-Cells

Define a Voronoi cell Ω(v) around a vertex v ∈ Λ is the set of points closer to v than to any other vertex. A set of triangles B in the packing, the triangles in this set are "barriers". A triangle with vertices in the packing belongs to B iff it's quasi-regular or the triangle is a face of a simplex in the Q-system. Note, that barriers can't overlap. A point y is obstructed at x ∈ ℝ³ if the line segment from x to y passes through the interior of a triangular region in B. 'Obstruction' is a symmetric relation. A saturated packing Λ and a map ϕ: ℝ³' → Λ so that the image of x lies in Λ(x, 2√3). If x ∈ ℝ³', and Λ_x = {w ∈ Λ: x ∈ I_w ∧ w unobstructed at x}. If Λ_x = ∅, then ϕ(x) is the vertex closest to x. If it's nonempty, then it's the vertex of Λ_x closest to x. For v ∈ Λ, VC(v) is the closure of ϕ^-1(v) in ℝ³, and the V-cell at v. V-cells cover space, with interiors of distinct V-cells being disjoint. Each V-cell is the closure of its interior, as well as a finite union of nonoverlapping convex polyhedra. At most one face of a quarter Q has negative orientation. A polynomial χ(x₁, ..., x₆) = x₁x₄x₅ + x₁x₆x₄ + x₂x₆x₅ + x₂x₄x₅ + x₅x₃x₆ + x₃x₄x₆ - 2x₅x₆x₄ - x₁x²₄ - x₂x²₅ - x₃x²₆, with x_i = y_i² where the tuples of y_i are the lengths of the edges of a simplex. A simplex S(y_i) has negative orientation along the face indexed by (4,5, 6) iff χ(y_i²) < 0.

A set of three vertices F with one edge between pairs of vertices has length between 2t₀ and √8 with the others at most 2t₀, then with v not on Q, if the simplex (F, v) has negative orientation along F, it's a quarter. Similarly, for a quasi-regular triangle, with some outside vertex v, the combined simplex with negative orientation S is a quasi-regular tetrahedron with |v - v_i| < 2t₀. If x lies in the open Voronoi cell at the origin, but outside the V-cell at origin, there is a simplex Q ∈ Q₀ such that x is in the cone over Q. x is outside the interior of Q.

For x ∈ I₀, the cube of side 4 centered at the origin parallel to coordinate axes and the closed segment {x, w} intersects the closed 2D cone with center 0 over F = {0, v₁, v₂}, F ∈ B'₀. If the origin is unobstructed at x, and x is closer to origin than v_i, then it's outside VC(w).