From:           asimov@nas.nasa.gov (Daniel A. Asimov)
Newsgroups:     sci.math.research
Subject:        Regular Polytopes in Hilbert Space
Date:           15 Aug 1995 22:18:33 GMT
Organization:   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.research
Subject:        Re: Regular Polytopes in Hilbert Space
Date:           16 Aug 1995 11:28:04 GMT
Organization:   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