From:asimov@nas.nasa.gov (Daniel A. Asimov)Newsgroups:sci.math.researchSubject:Regular Polytopes in Hilbert SpaceDate:15 Aug 1995 22:18:33 GMTOrganization:NAS - NASA Ames Research Center, Moffett Field, CA

Let H denote the Hilbert space of square-summable sequences of reals. QUESTION: What is the "right" definition of a regular polytope in H, and what is the classification of such things? Consider a bounded set S of points of H having no limit point in H. Then if the convex hull C(S) of S is to be called a regular polytope, I should think that it ought to satisfy, at minimum, the following condiitions: 1. C(S) is contained in no codimension-one affine subspace of H. 2. Let a "flag" be a sequence F = (F_0, F_1, F_2,...) of faces of C(S), where F_0 is a face of C(S) (codimension-one in H), and F_(k+1) is a codimension-one face of F_k for each k = 1,2,.... Then for any two such flags F' and F'', there exists an isometry of H carrying F' into F''. Two examples of such C(S) are the following: a) The convex hull of the set {e_i, -e_i : i = 1,2,...}, where e_i is the ith standard basis vector in H (1 in the ith place and 0 otherwise). (This seems analogous to the octahedron or "cross-polytope".) b) Let D = the convex hull of the set {e_i : 1 = 1,2,...}. This lies in the codimension-one affine subspace L = {(c_1,c_2,...) : sum c_i = 1}. Now let I: L -> H be an isometry of L onto H. Then I(D) satisfies 1. and 2. above. (This seems analogous to the tetrahedron or simplex.) (But there seems to be no analogue of the cube, since (1,1,...) is not in H.) QUESTION: Is there any example satisfying 1. and 2. that is essentially distinct from a) and b) ??? References to any literature on this subject would also be appreciated. Daniel Asimov Senior Computer Scientist Mail Stop T27A-1 NASA Ames Research Center Moffett Field, CA 94035-1000 asimov@nas.nasa.gov (415) 604-4799 w (415) 604-3957 fax

From:4jukka@adpser2.gsf.fi (Jukka Liukkonen)Newsgroups:sci.math.researchSubject:Re: Regular Polytopes in Hilbert SpaceDate:16 Aug 1995 11:28:04 GMTOrganization:Geological Survey of Finland

Daniel A. Asimov (asimov@nas.nasa.gov) wrote: : Let H denote the Hilbert space of square-summable sequences of reals. [snip] : (But there seems to be no analogue of the cube, since (1,1,...) is not in H.) [snip] Why should (1,1,...) belong to the cube? I suggest that the cube is the convex hull of the points (d_1, d_2, d_3, ...), where d_i \in {0, 1} \forall i = 1,2,... and d_i = 1 for only finite number of indices i. ---------------------------------------------------------------------- Jukka Liukkonen Geological Survey of Finland Senior ADP Designer Betonimiehenkuja 4 Internet: Jukka.Liukkonen@gsf.fi 02150 ESPOO Fax: 358-0-462205 FINLAND WWW: http://www.gsf.fi/~4jukka/ ----------------------------------------------------------------------