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ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow­ ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera­ lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) . Page 222. A very elegant, non-computational, construction of the 3-diagram E0 ' of theorem 11.5.2 was communicated to the author by G. Wegner. His construction is explained in an addendum to Griinbaum­ Sreedharan [1]. Concerning the dual A* of the 3-complex A represented by E0' (see page 224), Wegner has shown that it is not representable by a 3-diagram if the basis of the diagram is required to be the 3-face of A* which corresponds to the vertex 8 of A. Page 231, line 4. M. A. Perles has shown (private communication) that the graphs in question are dimensionally ambiguous whenever d ~ n + 3. 426 ADDEND UM 427 Page 271. In a revised version of Barnette's paper [3], the following improvement of the inequalities of exercise 13.3.13 is established: If (Pk) is a 3-realizable sequence, then 2P3 + 2P4 + 2ps + 2P6 + P7 ~ 16 + L (k - 8)Pk' k ?: 9 Thi s inequality refutes conjecture 2 (page 268). Indeed, if Pk = 0 for k -:f 3, 6, 6m, then equation (*) (page 254) implies P3 = 4 + (2m - 2)P6m' hence Barnette's inequality yields P6 ~ 4 + (m - 2)P 6m ' For (m - 2)(P6m - 6) ~ 8 this contradicts conjecture 2. On the other hand, an affi rmative solution of conjecture 1 (page 267) will be established in a forth coming paper by the author. Page 315. Recent results have greatly increased the number of known Euler-type relations, and their comparison here seems useful. We have alre ady discu ssed Euler's equation (Chapter 8) concerning numbers of faces, Gram's equation (theorem 14.1.1) concerning angle sums, and the equation of theorem 14.3.2 dealing with the Steiner point of a polytope and its faces. In order to discuss the new results, let pd = P1 denote a d-polytope, let Jj = HP), and let the j-faces of pdbe p{, ..., Pir Let m(K) denote the mean width of the compact convex set K. Shephard [11] proved that for each d-polytope P d I j L (-IY L m(P{) = -m(P). j;O j; 1 Another function with similar properties is the "angle-deficiency" l5(pL P) defined below. Let us define qJ(PL pt -I) as the angle (see page 297) spanned by pi in pt -I if pi c Pi: '; and as 0 if Pi is not contained in pt - I. Extending the well known case (d = 3), the following results were obtained by Perles-Shephard [1], Perles-Walkup [1], and Shephard [10]: For eachj-face pi ofthe d-polytope P (0::::; j ::::; d - 1), Id -I l5(pL P) = I- L qJ(P{, pt - I) ~ 0, j ; 1 with equality for j ~ d - 2, and strict inequality for j ::::; d - 3. For each d-polytope P, d -3 Ij L ( -lY L (j(P{, P) = 1 + (_I)d- I. j ; O j ; 1 428 CONVEX POLYTOPES An additional relation of the same type (which contains the mean width result as a special case) is due to Shephard [13]. Let p d be ad-polytope, let 0 < r < d, and let K,+ 1> "" K, be any d - r convex bodies. Let us denote by v(P{, . , PI, K,+ I' .. ., K d) the mixed volume (Bonnesen­ Fenchel [1], page 38) of the face p{ of pd (repeated r times) and the sets K,+ I> ••• , K d • Then d r, L (-IY L v(P{, .. ,P{,K'+I, · .. K d) j=O i =O = (-1)'v( -pd, ..., -r-, K,+I"" K d )· For proofs of these results, which usc a great variety of methods, the reader should consult the papers quoted. They contain also extensions to the case of spherical polytopes, as well as analogues of the Dehn-Sommer­ ville equations (in case of simplicial polytopes, and of other special families of polytopes) for the various quantities considered. One indication of the usefulness of some of these results may be found in Perles-Shephard [2]. Thi s paper contains results of the following type (which were completely unassailable so far) ; we quote only two very special results which are easy to formulate : If d ~ 7, no d-polytope has all facets combinatorially equ ivalent to the (d - 1)-octahedron. No 5-polytope has all facets combinatorially equivalent to the cyclic polytope C(8,4). Page 365. The example (figure 17.1.9) of a 3-valent, 3-connected, cyclically 5-connected planar graph without a Hamiltonian circuit (Walther [1]) was recently improved in some respects. While Walther's example contains 162 nodes, the author has found a similar example with 154 nodes, as well as an example with 464 nodes which does not admit even a Hamiltonian path. The first example was improved still further by Walther, who constructed a 114-node graph of this type which has no Hamiltonian circuit (private communication). A still smaller graph with the same property (with 46 nodes only) has reportedly been found by Kozyrev. Page 425. The coefficients of the Dehn-Sommerville equations given in table 3 were independently computed by Riordan [1]; however, his table is marred by misprints. In Riordan's notation, the incorrect values are Ai,_1(4) and Aj,2(5). ERRATA FOR THE 1967 EDITION This is a list of corrections (other than small typos), noted by Marge Bayer, Branko Grtinbaum, Michael Joswig, Volker Kaibel, Victor Klee, Carsten Lange, Julian Pfeifle, and GUnterM. Ziegler. A minus in front of a line number means "counted from the bottom" . Page Line Original Correction 9 6 Use exercise 2 Use statement 2 9 13 The relative interior of a convex set A C Rd may be defined as relintA := {x E Rd : (affA) n (x+ eB") C A for some e > OJ. This is empty if and only if A = 0. 13 20 The claim in exercise 7 is not valid. The following counter­ example is due to P. McMullen: Let K := K' + B3• where 3 K' := {(x,y,O) E R3 I x> 0, xy ~ I} and B3 := {(x,y,z) ER 1 x 2 +r + z2 ~ I} is the 3-dimensional unit ball. and let L := {(x, 0, I) Ix E R}. (An earlier counterexample. due to T. Botts. ap­ pears on p. 459 of Klee [a].) 35 - 15 P = conv(FUF1) aff(P) = aff(FUF)) 35 - I4 The hypothesis in exercise 3 should include Fk _) C FH) . 59 12 section 9.3 section 9.4 61 I 2d -) 2d - i 62 5 O~i~[id ] O~i<lid] 65 -5 K~ -K~ 68 15 P{Vj ) P). (V) 68 -3 9"(K) 9"(P) 77 16 AK AP 78 - II In part (ii) ofTheorem I the "only if' part is not true. Let F be a vertex of a polygon P and let V be the new vertex so that E' := conv({V} U F) is an edge that contains an original edge E of P. Then E' is a face of P·. but (*) is not satisfied. since V ~ aff F. and (**) is not satisfied. because there is no facet (edge) of P containing F for which V is beyond. The (first) error in the proof occurs in lines 4-5 on page 79. See Altshuler-Shemer [a]. 79 -7 aff FoC affF is false for the same reason as above. 82 17 The claim in exercise 13(ii) was not established by Shephard in [7]; he later found an error in his construction. (See also the notes in sec­ tion 4.9.) 100 - 8 [id2J [!d2] II 3 3 The last product in this formula contains two typos; it should be: TI Pj-l .28 je{ilr,<a,} Pj 428a 428b CONVEX POLYTOPES Page Line Original Correction lIS 0 In the figure, the star diagram and the Gale-diagram do not fit together (there should be a * at position (2,3». 117 10 el' e2,e3,e4 e)le2,e3,-e4 129 -I exercise 7.3.4 exercise 7.3.5 129 -3 exercise 7.3.4 exercise 7.3.5 138 3 section 3.3 section 3.2 170 4 Perles and Shephard did not prove the existence, for each d ~ 4, of infinitely many d-polytopes of type (2,d - 2).
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