1,730 research outputs found
Convex Hulls, Oracles, and Homology
This paper presents a new algorithm for the convex hull problem, which is
based on a reduction to a combinatorial decision problem
POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a
simplicial homology computation. Like other convex hull algorithms, our
algorithm is polynomial (in the size of input plus output) for simplicial or
simple input. We show that the ``no''-case of
POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in
polynomial time (if integrity of the input is guaranteed).Comment: 11 pages, 2 figure
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
On moments of a polytope
We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a compact
polytope P in R^d is a rational function. Its denominator is the product of
linear forms dual to the vertices of P raised to the power equal to the degree
of the density function. Using this, we solve the inverse moment problem for
the set of, not necessarily convex, polytopes having a given set S of vertices.
Under a weak non-degeneracy assumption we also show that the uniform measure
supported on any such polytope is a linear combination of uniform measures
supported on simplices with vertices in S.Comment: 28 pages, 3 figure
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
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