✍ Scribed by Magelone Kömhoff; G. C. Shephard
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ISOMORPHISM CLASSES OF CONVEX POLYTOPES
Let ~a denote the set of all d-dimensional convex polytopes (d-polytopes) in d-dimensional Euclidean space R d. For any Pe~ a we write [P] for the class of all d-polytopes combinatorially isomorphic to P. For the definitions of these terms, general information about convex polytopes, and explanations of standard terminology and notations, see [4] or [6].
For any polytopes P, Qe~ a we say that the class [Q] approximates the polytope P, and write [Q]~P, if there exists a sequence (Qi)•=l, with Qi~[Q], which converges to P in the Hausdorff metric. The problem of finding, in terms of P, necessary and sufficient conditions on [Q] for [Q] -~P seems to be intractable at present. Some of the difficulty arises from the fact (which is a consequence of our main theorem) that a solution must, in general, depend on P itself and not only on the combinatorial type of P. So far as we are aware, only one result in this direction has been published, namely that of Eggleston et aL [3], who showed that a necessary condition for [Q] ~P is that, for each dimension k=0 ..... d-1, the d-polytope Q must have at least as many k-dimensional faces (k-faces) as the d-polytope P.
If [Q] ~P implies that [Q] also approximates any polytope combinatorially isomorphic to P, then we shall write [Q] ~ [P]. The purpose of this paper will be to investigate under what conditions [Q] ~P implies [Q] ~ [P]. Our results are as follows. MAIN THEOREM. A. [Q] ~ P implies [Q] ~ [P] in at least the following cases. O) P, Q e~3, (ii) P, Qe~ a and either P is simple and Q has the same number of facets ((d-1)-faces) as P, or P is simplicial and Q has the same number of vertices as P, (iii) P, Qe~ a are non-pyramidal and both have at most d+3 vertices, or both have at most d + 3 facets. B. For each d>>,4 there exist non-pyramidal polytopes P, Qe~ a, each with d+4 or more vertices, or each with d+4 or more facets, such that [a] ~ P does not imply that [Q] ~ [P].
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