7 research outputs found
Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements
Monotone path polytopes arise as a special case of the construction of fiber
polytopes, introduced by Billera and Sturmfels. A simple example is provided by
the permutahedron, which is a monotone path polytope of the standard unit cube.
The permutahedron is the zonotope polar to the braid arrangement. We show how
the zonotopes polar to the cones of certain deformations of the braid
arrangement can be realized as monotone path polytopes. The construction is an
extension of that of the permutahedron and yields interesting connections
between enumerative combinatorics of hyperplane arrangements and geometry of
monotone path polytopes
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
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version