Projective Planes by D.R. Hughes

By D.R. Hughes

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Then the packing P is uniformly stable. 1. 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. Qi / p n. 3 Proof. 18) where 1 Ä i Ä n. (For a proof we refer the interested reader to p. 21) holds for all 1 Ä i Ä n. Now, let s C be a closed line segment along which exactly k members of the family fQ1 ; Q2 ; : : : ; Qn g meet having inner dihedral angles ˇ1 ; ˇ2 ; : : : ; ˇk . There are the following three possibilities: (a) s is on an edge of the cube C; (b) s is in the relative interior either of a face of C or of a face of a convex cell in the family fQ1 ; Q2 ; : : : ; Qn g; (c) s is in the interior of C and not in the relative interior of any face of any convex cell in the family fQ1 ; Q2 ; : : : ; Qn g.

See [159] and [160] for the original version of the lemma, which is somewhat different from the equivalent version below. 3. , let W be a d -dimensional orthoscheme in Ed ). o; convfui ; ui C1 ; : : : ; ud g/ for all 1 Ä i Ä d . B \ U/ where B stands for the d -dimensional unit ball centered at the origin o of Ed . 3 with the additional property that kwi k D 2i i C1 for all 1 Ä i Ä d . d C 1/Š d -dimensional simplices, each congruent to W. 16) Finally, let U WD Vf D convfo; v1 ; v2 ; : : : ; vd g for Vf 2 V.

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