843 research outputs found
The -Construction for Lattices, Spheres and Polytopes
We describe and analyze a new construction that produces new Eulerian
lattices from old ones. It specializes to a construction that produces new
strongly regular cellular spheres (whose face lattices are Eulerian). The
construction does not always specialize to convex polytopes; however, in a
number of cases where we can realize it, it produces interesting classes of
polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple
4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct
for each an infinite family of -simplicial 2-simple
-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure
Bier spheres and posets
In 1992 Thomas Bier presented a strikingly simple method to produce a huge
number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a
simplicial complex on n vertices with its combinatorial Alexander dual.
Here we interpret his construction as giving the poset of all the intervals
in a boolean algebra that "cut across an ideal." Thus we arrive at a
substantial generalization of Bier's construction: the Bier posets Bier(P,I) of
an arbitrary bounded poset P of finite length. In the case of face posets of PL
spheres this yields cellular "generalized Bier spheres." In the case of
Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I)
inherit these properties.
In the boolean case originally considered by Bier, we show that all the
spheres produced by his construction are shellable, which yields "many
shellable spheres", most of which lack convex realization. Finally, we present
simple explicit formulas for the g-vectors of these simplicial spheres and
verify that they satisfy a strong form of the g-conjecture for spheres.Comment: 15 pages. Revised and slightly extended version; last section
rewritte
Inhomogeneous extreme forms
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two
reduction theories for lattices, well-suited for sphere packing and covering
problems. In his first memoir a characterization of locally most economic
packings is given, but a corresponding result for coverings has been missing.
In this paper we bridge the two classical memoirs.
By looking at the covering problem from a different perspective, we discover
the missing analogue. Instead of trying to find lattices giving economical
coverings we consider lattices giving, at least locally, very uneconomical
ones. We classify local covering maxima up to dimension 6 and prove their
existence in all dimensions beyond.
New phenomena arise: Many highly symmetric lattices turn out to give
uneconomical coverings; the covering density function is not a topological
Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in
Annales de l'Institut Fourie
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