By Peter McMullen, Egon Schulte

Summary usual polytopes stand on the finish of greater than millennia of geometrical study, which begun with usual polygons and polyhedra. The speedy improvement of the topic some time past two decades has led to a wealthy new conception that includes an enticing interaction of mathematical components, together with geometry, combinatorics, staff conception and topology. this is often the 1st accomplished, up to date account of the topic and its ramifications. It meets a severe want for this type of textual content, simply because no publication has been released during this quarter when you consider that Coxeter's "Regular Polytopes" (1948) and "Regular advanced Polytopes" (1974).

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Since F ⊂ C, f F = F . Then f D has f F = F as facet in common with f C = C. By the thin-ness of X, this means that f D is either D or C, since those are the only two chambers with facet F . Since f D = D, we have f D = C. ♣ Proposition: The half-apartment Φ is convex in the sense that, given C, D both in Φ, every minimal gallery γ = Co , . . , Cn connecting C, D lies entirely inside Φ. Proof: Let γ = Co , . . , Cn be a minimal gallery connecting C, D. Suppose that some Ci does not lie in Φ. Then there is i so that Ci ∈ Φ but Ci+1 ∈ Φ.

Fix a chamber C in X. Let f : X → Y , g : X → Y be chamber complex maps which agree pointwise on C, and both of which send non-stuttering galleries (starting at C) to non-stuttering galleries. Then f = g. Proof: Let γ be a non-stuttering gallery C = C0 , C1 , . . , Cn = D. By hypothesis, f γ and gγ do not stutter. That is, f Ci = f Ci+1 for all i, and similarly for g. Suppose, inductively, that f agrees with g pointwise on Ci . Certainly f Ci and f Ci+1 are adjacent along F = f Ci ∩ f Ci+1 = gCi ∩ gCi+1 By the non-stuttering assumption, f Ci+1 = f Ci and gCi+1 = gCi .

Cn = D connecting C and D. We will use this notation for the following lemmas. Lemma: There exist adjacent chambers C, D so that f C = C and f D = D. For any such C, D, we have f D = C. Therefore, if γ is a gallery from A to B with f A = A and f B = B, then f γ must stutter. Proof: There are chambers A, B so that f A = A and f B = B, by definition of a folding. There is a gallery A = Co , . . , Cn = B connecting the two, so there is a least index i so that f Ci = Ci and f Ci+1 = Ci+1 . Take C = Ci and D = Ci+1 .