Multiple Covers with Balls II: Weighted Averages

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

“…), as proved in [5], which implies that Del(A k,1 , w k,1 ) is an orthogonal dual of Bri k (A); see the lower middle panel in Figure 1. We remark that [5] describes a 1-parameter family of coefficients that generate points with real weights whose weighted order-1 Voronoi tessellations are the order-k Brillouin tessellation of A.…”

“…), as proved in [5], which implies that Del(A k,1 , w k,1 ) is an orthogonal dual of Bri k (A); see the lower middle panel in Figure 1. We remark that [5] describes a 1-parameter family of coefficients that generate points with real weights whose weighted order-1 Voronoi tessellations are the order-k Brillouin tessellation of A. In particular, there are two positive coefficients, w 1 < w 0 , that satisfy (k − 1)w 0 + w 1 = 1, and for every B ⊆ A of size k and b ∈ B, we use w 1 for b and w 0 for every other point in B.…”

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

“…), as proved in [5], which implies that Del(A k,1 , w k,1 ) is an orthogonal dual of Bri k (A); see the lower middle panel in Fig. 1.…”

“…We remark that [5] describes a 1-parameter family of coefficients that generate points with real weights whose weighted order-1 Voronoi tessellations are the order-k Brillouin tessellation of A. In particular, there are two positive coefficients, w 1 < w 0 , that satisfy (k − 1)w 0 + w 1 = 1, and for every B ⊆ A of size k and b ∈ B, we use w 1 for b and w 0 for every other point in B.…”

“…1. Definition 2.4 (orthogonal dual mosaics) Noting that they are orthogonal duals of Vor k (A) and Bri k (A), we call Del k (A) := Del(A k , w k ) the order-k Delaunay mosaic and Igl k (A) := Del(A k,1 , w k,1 ) the order-k Iglesias mosaic of A; see [5].…”