Long monotone paths on simple 4-polytopes

Details Long monotone paths on simple 4-polytopes Long monotone paths on simple 4-polytopes

2004-02-15

arxiv.org/abs/math/0402247v1

Mathematics

Metric Geometry

19 pages, 12 figures

Scientific paper

The Monotone Upper Bound Problem (Klee, 1965) asks if the number M(d,n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d,n) = M_{ubt}(d,n), the maximal number of vertices that a d-polytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d=4, the answer is ``yes'', despite the fact that it is ``no'' if we restrict ourselves to the dual-to-cyclic polytopes. For each n>=5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope with n facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function. This constrasts an earlier result, by which no polar-to-neighborly 6-dimensional polytope with 9 facets admits a monotone Hamilton path.

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