441 research outputs found
Fiber polytopes for the projections between cyclic polytopes
The cyclic polytope is the convex hull of any points on the
moment curve in . For , we
consider the fiber polytope (in the sense of Billera and Sturmfels) associated
to the natural projection of cyclic polytopes which
"forgets" the last coordinates. It is known that this fiber polytope has
face lattice indexed by the coherent polytopal subdivisions of which
are induced by the map . Our main result characterizes the triples
for which the fiber polytope is canonical in either of the following
two senses:
- all polytopal subdivisions induced by are coherent,
- the structure of the fiber polytope does not depend upon the choice of
points on the moment curve.
We also discuss a new instance with a positive answer to the Generalized
Baues Problem, namely that of a projection where has only
regular subdivisions and has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur
Linear inequalities for flags in graded posets
The closure of the convex cone generated by all flag -vectors of graded
posets is shown to be polyhedral. In particular, we give the facet inequalities
to the polar cone of all nonnegative chain-enumeration functionals on this
class of posets. These are in one-to-one correspondence with antichains of
intervals on the set of ranks and thus are counted by Catalan numbers.
Furthermore, we prove that the convolution operation introduced by Kalai
assigns extreme rays to pairs of extreme rays in most cases. We describe the
strongest possible inequalities for graded posets of rank at most 5
Splines in geometry and topology
This survey paper describes the role of splines in geometry and topology,
emphasizing both similarities and differences from the classical treatment of
splines. The exposition is non-technical and contains many examples, with
references to more thorough treatments of the subject.Comment: 18 page
A better upper bound on the number of triangulations of a planar point set
We show that a point set of cardinality in the plane cannot be the vertex
set of more than straight-edge triangulations of its convex
hull. This improves the previous upper bound of .Comment: 6 pages, 1 figur
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