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1.
Luis Paris 《Geometriae Dedicata》1993,45(2):171-176
We denote by the complement of the complexification of a real arrangement of hyperplanes. It is known that there is a certain technical property, called property D, on real arrangements of hyperplanes such that: if a real arrangement of hyperplanes is simplicial then has property D, and if has property D then is aK(, 1) space. Our main goal is to prove that: if has property D then is simplicial. We also prove that a quasi-simplicial arrangement is always simplicial. 相似文献
2.
Arrangements of hyperplanes and Lie algebra homology 总被引:6,自引:0,他引:6
3.
Sol Schwartzman 《Proceedings of the American Mathematical Society》2002,130(5):1457-1458
Let be a smooth strictly convex closed hypersurface in and let be any oriented smooth connected manifold immersed in Suppose that is a continuous function from to Then there is at least one point such that the hyperplane tangent to at is parallel to the hyperplane tangent to the immersed manifold at the point corresponding to If there did not exist at least two such points, would have to be compact and the Hurewicz homomorphism of into would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of would have to be equal to For any and any immersed we could always get maps for which the number of points satisfying the conditions of our theorem exactly equaled two. An example can be given in which both and are the unit sphere about the origin in and there is only one such point .
4.
M. Edelstein 《Israel Journal of Mathematics》1969,7(1):90-94
LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty).
This research was supported by the National Research Council of Canada, Grant A-3999. 相似文献
5.
We call a convex subsetN of a convexd-polytopePE
d
ak-nucleus ofP ifN meets everyk-face ofP, where 0<k<d. We note thatP has disjointk-nuclei if and only if there exists a hyperplane inE
d
which bisects the (relative) interior of everyk-face ofP, and that this is possible only if [d+2/2]kd–1. Our main results are that any convexd-polytope with at most 2d–1 vertices (d3) possesses disjoint (d–1)-nuclei and that 2d–1 is the largest possible number with this property. Furthermore, every convexd-polytope with at most 2d facets (d3) possesses disjoint (d–1)-nuclei, 2d cannot be replaced by 2d+2, and ford=3, six cannot be replaced by seven.Partially supported by Hung. Nat. Found. for Sci. Research number 1238.Partially supported by the Natural Sciences and Engineering Council of Canada.Partially supported by N.S.F. grant number MCS-790251. 相似文献
6.
Rolf Schneider 《Discrete and Computational Geometry》1987,2(1):223-232
Let be a locally finite system of hyperplanes in
d
with the property that the cells of the induced cell complex decomposition of
d
have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained. 相似文献
7.
Bernard Chazelle 《Discrete and Computational Geometry》1993,9(1):145-158
Givenn hyperplanes inE
d, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverE
d and such that the interior of each simplex intersects at mostn/r hyperplanes. We present a deterministic algorithm for computing a (1/r)-cutting ofO(r
d) size inO(nr
d–1) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.This research was supported in part by the National Science Foundation under Grant CCR-9002352. 相似文献
8.
Takuro Abe 《Inventiones Mathematicae》2016,204(1):317-346
9.
J. Arias de Reyna 《Mathematische Annalen》1980,249(2):111-114
10.
Thomas Zaslavsky 《Geometriae Dedicata》1983,14(3):243-259
We show how the results of Dowling and Wilson on Whitney numbers in ‘The slimmest geometric lattices’ imply minimum values for the numbers of k-dimensional flats and d-dimensional cells of a projective d-arrangement of hyperplanes and for the number of d-cells missed by an extra hyperplane. Their theorems also characterize the extremal arrangements. We extend their lattice results to doubly indexed Whitney numbers; thence we obtain minima for the number of k-dimensional cells and the number of pairs of flats with x \(\subseteq\) y and dim x=k, dim y=l. The lower bounds are in terms of the rank and number of points of the geometric lattice, or the dimension d and the number of hyperplanes of the arrangement. The minima for k-cells were conjectured by Grünbaum; R. W. Shannon proved the minima for k-dimensional flats and cells and characterized attainment for the latter by a more strictly geometric, non-latticial technique. 相似文献
11.
Any sequence of points in Rn obtained by successive projections of a point on elements of a finite set of hyperplanes is bounded. 相似文献
12.
《Optimization》2012,61(7):1409-1438
Every pair of relatively disjoint polytopes is dual to the parameter space of all their separating hyperplanes, which is also a polytope. For a polytope whose interior is disjoint from the relative interior of another polytope, the parameter space of all separating hyperplanes is a polytope of the same dimension. One face of this parameter space parametrizes the separating hyperplanes that also simultaneously support both polytopes. A separating hyperplane corresponds to a vertex of this face if and only if no other hyperplanes support the polytopes at the same intersection points. If all the vertices of the polytopes have all their coordinates in an ordered field, then the same results and their proofs hold with the same ordered field. 相似文献
13.
V. Nitica 《Linear algebra and its applications》2010,432(1):402-1597
In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form
a1⊗x1⊕…⊕an⊗xn⊕an+1=b1⊗x1⊕…⊕bn⊗xn⊕bn+1, 相似文献
14.
William A. Arvola 《manuscripta mathematica》1991,71(1):295-306
LetV be a finite dimensional vector space over the real or complex numbers. Areal (orcomplex)arrangement
A inV is a finite collection of real (or complex) affine hyperplanes. A real arrangement inV can becomplexified to form a complex arrangement in the complex vector spaceA. The (complex)complement of a real arrangementA is defined byM(A)=V⊗ℂ−⋃
H∈
A
H⊗ℂ. There are two different finite simplicial complexes which carry the homotopy type ofM(A), one given by M. Salvetti, the other by P. Orlik. In this paper we describe both complexes and exhibit a simplicial homotopy
equivalence between them. 相似文献
15.
16.
Luc Teirlinck 《Journal of Combinatorial Theory, Series A》1980,28(3):290-306
We study projective and affine factors in 2-coverings and associate canonically a projective space of order 2 and an affine space of order 3 to each Steiner triple system and an affine space of order 2 to each Steiner quadruple system. 相似文献
17.
18.
Hyperplane arrangements of rank 3 admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove that arrangements which admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary we give a new proof that irreducible arrangements with a generic hyperplane are totally nonfree. In another application we show that an irreducible multiarrangement of rank 3 with at least two locally heavy hyperplanes is not free. 相似文献
19.
E. A. Ramos 《Discrete and Computational Geometry》1996,15(2):147-167
We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR
d
, there arek hyperplanes so that each orthant contains a fraction 1/2
k
of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2
k
−1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2
k−1. We are able to prove that Δ(j, k)=⌈j(2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2
n
withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove
a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5).
This research was supported by the National Science Foundation under Grant CCR-9118874. 相似文献
20.
For a 3-connected binary matroid M, let dimA(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dimA(M)=r(M). In 2004, when |A|=1, Lemos proved that dimA(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dimA(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid. 相似文献