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1.
A characterization theorem is given for 3-dimensional convex polytopes Q having the following property: There exists a polytope P, isomorphic to Q, all edges of which can be cut by a pair of planes that miss all its vertices. The result yields an affirmative solution of a conjecture of B. Grünbaum. 相似文献
2.
In this paper we relate the theory of stable planes to the theory of generalized symmetric spaces in the sense of differential geometry where the symmetries may be of arbitrary order. This leads to the notion of a generalized symmetric plane. We assign to every generalized symmetric plane an associated infinitesimal model and show that the associated infinitesimal model essentially determines a generalized symmetric plane up to global isomorphism. In particular, every generalized symmetric plane with an abelian group of transvections is a topological translation plane. 相似文献
3.
An algorithm for generating cutting planes for mixed-integer knapsack polyhedra is described. The algorithm represents an exact separation procedure and is based on a general methodology proposed by one of the authors in an earlier paper. Computational results are presented. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. 相似文献
5.
Summary In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regular n-gon, n 5, is cut into m triangles of equal areas, then m is a multiple of n. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction regular is deleted from the hypothesis and prove that it does for n = 6 and n = 8. 相似文献
6.
It is shown that if, for some 2≦ j≦ d-2, all the j-faces of a d-polytope P are centrally symmetric, then all the faces of P of every dimension are centrally symmetric. 相似文献
7.
Let K be a three-dimensional centrally symmetric compact convex set of unit volume. It is proved that K is contained in a centrally symmetric hexagonal prism (or a parallelepiped) of volume 4 | / |
3?{3} < 2.7734451 {{4} \left/ {{\sqrt[3]{3} < 2.7734451}} \right.} . This fact implies that space contains a lattice packing of translates of K with density $ {{{\sqrt[3]{3}}} \left/ {{4 > 0.36056}} \right.} $ {{{\sqrt[3]{3}}} \left/ {{4 > 0.36056}} \right.} . Furthermore, K is contained in a parallelepiped of volume
\frac43( 2 + ?3 )2 | / |
3 < 3.2080203 \frac{4}{3}{\left( {2 + \sqrt {3} } \right)^{{{2} \left/ {3} \right.}}} < 3.2080203 . Bibliography: 6 titles. 相似文献
8.
A new and conceptually simpler proof is given of the theorem of A. D. Aleksandrov and G. C. Shephard, that a d-polytope ( d≧3), all of whose facets are centrally symmetric, is itself centrally symmetric. 相似文献
9.
Bambah and Rogers proved that the area of a convex domain in the plane which can be covered by n translates of a given centrally symmetric convex domain C is at most ( n–1) h( C)+ a( C), where h( C) denotes the area of the largest hexagon contained in C and a( C) stands for the area of C. An improvement with a term of magnitude n is given here. Our estimate implies that if C is not a parallelogram, then any covering of any convex domain by at least 26 translates of C is less economic than the thinnest covering of the whole plane by translates of C. 相似文献
10.
The note contains an example of three plane convex centrally symmetric figures P 1, P 2, P 3 such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent to P 1, P 2, P 3 respectively. 相似文献
12.
We prove tight lower bounds for the coefficients of the toric h-vector of an arbitrary centrally symmetric polytope generalizing previous results due to R. Stanley and the author using
toric varieties. Our proof here is based on the theory of combinatorial intersection cohomology for normal fans of polytopes
developed by G. Barthel, J.-P. Brasselet, K. Fieseler and L. Kaup, and independently by P. Bressler and V. Lunts. This theory
is also valid for nonrational polytopes when there is no standard correspondence with toric varieties. In this way we can
establish our bounds for centrally symmetric polytopes even without requiring them to be rational.
Received: 24 March 2004 相似文献
13.
Geometriae Dedicata - We estimate the bottom of the $$L^2$$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincaré series. Our main... 相似文献
15.
Recently A. Dress completed the classification of the regular polyhedra in E
3 by adding one class to the enumeration given by Grünbaum on this subject. This classification is the only systematic study of a collection of polyhedra possessing special symmetries which uses the generalized definition of a polygon allowing for skew polygons as well as planar polygons in E
3. This study gives necessary conditions for polyhedra to be vertex-transitive and edge-transitive. These conditions are restrictive enough to make the task of completely enumerating such polyhedra realizable and efficient. Examples of this process are given, and an explanation of the basic process is discussed. These new polyhedra are appearing more frequently in applications of geometry, and this examination is a beginning of the classifications of polyhedra having special symmetries even though there are many other such classes which lack this scrutiny. 相似文献
16.
We show that the graphs of the centrally symmetric 3-polytopes can be generated from the graphs of the cube and octahedron by applying pairs of symmetric face splittings. 相似文献
17.
The conjecture that among convex bodies Q in R n, with a center of symmetry at the origin, for which
, the value of
is a maximum when Q is the layer between two hyperplanes, is proved for n=2 and n=3. Various approaches to the problem are
discussed as well as related unsolved problems.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad.
Nauk SSSR, Vol. 45, pp. 75–82, 1974. 相似文献
18.
We determine the class of all locally compact stable planes M of positive dimension d4 which admit a reflection at each point of some open set U
M. Apart from the expected possibilities (planes defined by real and complex hermitian forms, and almost projective translation planes), one obtains (subplanes of) H. Salzmann's modified real hyperbolic planes [14; 5.3] and one exceptional plane which was not known before. The case U=M has been treated [9] and is reproved here in a simpler way. The solution to the problem indicated in the title constitutes the main step in the proof of our results. 相似文献
19.
We investigate the incidence matrix of a finite plane of order n which admits a ( C, L)-transitivity G. The elation group G affords a generalized Hadamard matrix H=(h
ij
) of order n and an incidence matrix for the plane is completely determined by the matrix R(H)=(R(h
ij
)), where R(g) denotes the regular permutation matrix for gG. We prove that in the case R(H) is symmetric that G is an elementary abelian 2-group or else G is a nonabelian group and n is a square. Results are obtained in the abelian case linking the roots of the incidence matrix R(H) to the roots of the complex matrix ( H), a nontrivial character of G. 相似文献
20.
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of a centrally symmetric convex body. Our main tool is a generalization of a result of Davenport that bounds the number of lattice points in terms of volumes of suitable projections. 相似文献
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