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1.
Schneider 《Discrete and Computational Geometry》2003,29(4):575-593
Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies.
By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices
and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes
with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes
simplices. 相似文献
2.
With a slight modification of the original definition by Billera and Sturmfels, it
is shown that the theory of fibre polytopes extends to one of mixed fibre
polytopes. Indeed, there is a natural surjective homomorphism from the space of
tensor weights on polytopes in a euclidean space V to the corresponding space
of such weights on fibre polytopes in a subspace of V. Moreover, these
homomorphisms compose in the correct way; this is in contrast to the original
situation of the fibre polytope construction. 相似文献
3.
Annals of Combinatorics - The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over previous years. It is well known that a lattice polytope has... 相似文献
4.
Benjamin Matschke Julian Pfeifle Vincent Pilaud 《Discrete and Computational Geometry》2011,46(1):100-131
Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. 相似文献
5.
A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification
of the totally splittable polytopes. 相似文献
6.
A known characterization of the decomposability of polytopes is reformulated in a way which may be more computationally convenient,
and a more transparent proof is given. New sufficient conditions for indecomposability are then deduced, and illustrated with
some examples. 相似文献
7.
Neighborly cubical polytopes exist: for any n≥ d≥ 2r+2 , there is a cubical convex d -polytope C
d
n
whose r -skeleton is combinatorially equivalent to that of the n -dimensional cube. This solves a problem of Babson, Billera, and Chan.
Kalai conjectured that the boundary of a neighborly cubical polytope C
d
n
maximizes the f -vector among all cubical (d-1) -spheres with 2
n
vertices. While we show that this is true for polytopal spheres if n≤ d+1 , we also give a counterexample for d=4 and n=6 .
Further, the existence of neighborly cubical polytopes shows that the graph of the n -dimensional cube, where n\ge5 , is ``dimensionally ambiguous' in the sense of Grünbaum. We also show that the graph of the 5 -cube is ``strongly 4 -ambiguous.'
In the special case d=4 , neighborly cubical polytopes have f
3
=(f
0
/4) log
2
(f
0
/4) vertices, so the facet—vertex ratio f
3
/f
0
is not bounded; this solves a problem of Kalai, Perles, and Stanley studied by Jockusch.
Received December 30, 1998. Online publication May 15, 2000. 相似文献
8.
Consider an arrangement of n hyperplanes in \real
d
. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper.
We aim to bound their maximum combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes
are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the
past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for
d>2 . We analyze families of polytopes that do not share vertices. In \real
3
we show an O(k
1/3
n
2
) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results,
we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in \real
d
is Ω(k
1/2
n
d/2
) which is a factor of away from the best known upper bound in the range n
d-2
≤ k ≤ n
d
. The case where 1≤ k ≤ n
d-2
is completely resolved as a known Θ(kn) bound for cells applies here.
Received September 20, 1999, and in revised form March 10, 2000. Online publication September 22, 2000. 相似文献
9.
Moshe Rosenfeld 《Israel Journal of Mathematics》1975,21(1):24-30
A study is made of the non-regular planar 3-connected graphs with constant weight. 相似文献
10.
This paper is a study of the polyhedral geometry of Gelfand–Tsetlin
polytopes arising in the representation theory of ${\frak gl}_n \Bbb C$ and algebraic combinatorics.
We present a combinatorial characterization of the vertices and a method
to calculate the dimension of the lowest-dimensional face containing a given
Gelfand–Tsetlin pattern.
As an application, we disprove a conjecture of Berenstein and Kirillov
about the integrality of all vertices of the Gelfand–Tsetlin polytopes. We can
construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators
as $n$ grows, of a nonintegral vertex. This is the first infinite family of
nonintegral polyhedra for which the Ehrhart counting function is still a polynomial.
We also derive a bound on the denominators for the nonintegral vertices
when $n$ is fixed. 相似文献
11.
Tudor Zamfirescu 《Periodica Mathematica Hungarica》2008,57(2):227-230
Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question. 相似文献
12.
13.
14.
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class
of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two constructions of triangulations
of hypersimplices due to Stanley and Sturmfels and explain them in terms of alcoved polytopes. We study triangulations of
alcoved polytopes, the adjacency graphs of these triangulations, and give a combinatorial formula for volumes of these polytopes.
In particular, we study a class of matroid polytopes, which we call the multi-hypersimplices. 相似文献
15.
For reciprocation with respect to a sphere x2=c in Euclideann-space, there is a unitary analogue: Hermitian reciprocationwith respect to an antisphere u=c. This is now applied, forthe first time, to complex polytopes. When a regular polytope has a palindromic Schläfli symbol,it is self-reciprocal in the sense that its reciprocal ', withrespect to a suitable concentric sphere or antisphere, is congruentto . The present article reveals that and ' usually have togetherthe same vertices as a third polytope + and the same facet-hyperplanesas a fourth polytope (where + and are againregular), so as to form a compound, +[2].When the geometry is real, + is the convex hull of and ', while is their common content or core. For instance,when is a regular p-gon {p}, the compound is
The exceptions are of two kinds. In one, + and are notregular. The actual cases are when is an n-simplex {3, 3, ...,3} with n4 or the real 4-dimensional 24-cell {3, 4, 3}=2{3}2{4}2{3}2or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3.The other kind of exception arises when the vertices of arethe poles of its own facet-hyperplanes, so that , ', + and all coincide. Then is said to be strongly self-reciprocal. 相似文献
16.
René Brandenberg 《Discrete and Computational Geometry》2005,33(1):43-55
For the first time complete lists of two pairs of inner and outer
radii classes of the
three types of regular polytopes which exist in all dimensions are
presented. A new approach using isotropic polytopes provides better
understanding of the
underlying geometry and helps to unify the results. 相似文献
17.
M. I. Hartley 《Discrete and Computational Geometry》1999,21(2):289-298
In this paper it is shown that any (abstract) polytope is a quotient of a regular polytope by some subgroup N of the automorphism group W of , and also that isomorphic polytopes are quotients of by conjugate subgroups of W . This extends work published in 1980 by Kato, who proved these results for a restricted class of polytopes which he called ``regular'. The methods used here are more elementary, and treat the problem in a purely nongeometric manner. Received January 27, 1997, and in revised form October 1, 1997. 相似文献
18.
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their
integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons
is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes,
we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include
absolute irreducibility testing and factorization of polynomials via their Newton polytopes.
Received December 2, 1999, and in revised form November 6, 2000. Online publication May 4, 2001. 相似文献
19.
It is shown that the set [G,] of immersed linear networks in
that are parallel to a given immersed linear network
and have the same boundary as is a convex polyhedral subset of the configuration space of movable vertices of the graph G. The dimension of [G,] is calculated, and the number of its maximal faces is estimated. As an application, the spaces of all locally minimal and weighted minimal networks with fixed boundary and topology in
are described. Bibliography: 21 titles. 相似文献
20.
We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface \({\mathcal{H}}\) in \({\mathbb{C}^n}\) using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straight-line program, as a determinant, or as an oracle. The second algorithm assumes that \({\mathcal{H}}\) is represented numerically via a witness set. That is, it computes the Newton polytope of \({\mathcal{H}}\) using only the ability to compute numerical representatives of its intersections with lines. Such witness set representations are readily obtained when \({\mathcal{H}}\) is the image of a map or is a discriminant. We use the second algorithm to compute a face of the Newton polytope of the Lüroth invariant, as well as its restriction to that face. 相似文献