464 research outputs found
Every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces
We prove that every symplectic toric orbifold is a centered reduction of a
Cartesian product of weighted projective spaces. A theorem of Abreu and
Macarini shows that if the level set of the reduction passes through a
non-displaceable set then the image of this set in the reduced space is also
non-displaceable. Using this result we show that every symplectic toric
orbifold contains a non-displaceable fiber and we identify this fiber.Comment: 20 pages, 11 figures; Final version. Accepted at IMRN. Comments from
the referees included. Section about Gromov width added. Moreover we fixed
some small mistakes that unfortunately made it to the published version
(moment polytope for the weighted projective space was not fully correct; at
some point a not connected subgroup was called a torus
Geometric Reasoning with polymake
The mathematical software system polymake provides a wide range of functions
for convex polytopes, simplicial complexes, and other objects. A large part of
this paper is dedicated to a tutorial which exemplifies the usage. Later
sections include a survey of research results obtained with the help of
polymake so far and a short description of the technical background
Pseudograph associahedra
Given a simple graph G, the graph associahedron KG is a simple polytope whose
face poset is based on the connected subgraphs of G. This paper defines and
constructs graph associahedra in a general context, for pseudographs with loops
and multiple edges, which are also allowed to be disconnected. We then consider
deformations of pseudograph associahedra as their underlying graphs are altered
by edge contractions and edge deletions.Comment: 25 pages, 22 figure
Graphs of Transportation Polytopes
This paper discusses properties of the graphs of 2-way and 3-way
transportation polytopes, in particular, their possible numbers of vertices and
their diameters. Our main results include a quadratic bound on the diameter of
axial 3-way transportation polytopes and a catalogue of non-degenerate
transportation polytopes of small sizes. The catalogue disproves five
conjectures about these polyhedra stated in the monograph by Yemelichev et al.
(1984). It also allowed us to discover some new results. For example, we prove
that the number of vertices of an transportation polytope is a
multiple of the greatest common divisor of and .Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of
several lemmas and the upper bound in Theorem 1.1 is improved by a factor of
tw
Syntactic aspects of hypergraph polytopes
This paper introduces an inductively defined tree notation for all the faces
of polytopes arising from a simplex by truncations. This notation allows us to
view inclusion of faces as the process of contracting tree edges. Our notation
instantiates to the well-known notations for the faces of associahedra and
permutohedra. Various authors have independently introduced combinatorial tools
for describing such polytopes. We build on the particular approach developed by
Dosen and Petric, who used the formalism of hypergraphs to describe the
interval of polytopes from the simplex to the permutohedron. This interval was
further stretched by Petric to allow truncations of faces that are themselves
obtained by truncations, and iteratively so. Our notation applies to all these
polytopes. We illustrate this by showing that it instantiates to a notation for
the faces of the permutohedron-based associahedra, that consists of
parenthesised words with holes. Dosen and Petric have exhibited some families
of hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra)
describing the coherences, and the coherences between coherences etc., arising
by weakening sequential and parallel associativity of operadic composition. We
complement their work with a criterion allowing us to recover the information
whether edges of these "operadic polytopes" come from sequential, or from
parallel associativity. We also give alternative proofs for some of the
original results of Dosen and Petric.Comment: 42 pages, 4 figure
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