By Eisenhart L.P.

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**Example text**

4. ci C rB/ where vol3 . / refers to the 3-dimensional volume of the corresponding set. S Proof. ci C rO B/ into truncated Voronoi cells as follows. Let Pi denote the Voronoi cell of the packing P assigned to ci C B, 1 Ä i Ä n, that is, let Pi stand for the set of points of E3 that are not farther away from ci than from any other cj with j ¤ i; 1 Ä j Ä n. Then, recall the well-known fact (see for example, [99]) that the Voronoi cells Pi , 1 Ä i Ä n just introduced form a tiling of E3 . c C r O B/ for the packing P.

Let P WD convfp1 ; p2 ; : : : ; pn g be a d -dimensional convex polytope in Ed ; d 2 with vertices p1 ; p2 ; : : : ; pn . 39) Let F0 F1 Fl ; 0 Ä l Ä d 1 denote a sequence of faces, called a (partial) flag of P, where F0 is a vertex and Fi 1 is a facet (a face one dimension lower) of Fi for i D 1; : : : ; l. Then the simplices of the form convfcF0 ; cF1 ; : : : ; cFl g constitute a simplicial complex CP whose underlying space is the boundary of P. We regard all points in Ed as row vectors and use qT for the column vector that is the transpose of the row vector q.

1 2 / and Á p 3 2 /Or . cj C rO B/. 2) imply the following estimate. 8. ci C rO B/ < 24:53902 3 Proof. n m !! ci C rO B/ i D1 ! n 3 ! 6. 8)). i i / is based p on the new parameter value rN WD 2 (replacing rO D 1:81383). The details are as follows. First, recall that if f cc denotes the face-centered cubic lattice with shortest nonzero lattice vector of length 2 in E3 and we place unit balls centered at each lattice point of f cc , then we get the fcc lattice packing of unit balls, labelled by Pf cc , in which each unit ball is touched by 12 others such that their centers form the vertices of a cuboctahedron.