Intersecting all edges of convex polytopes by planes

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.     

Generalized symmetric planes

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.     

Cutting planes for mixed-integer knapsack polyhedra

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.     

Equidissections of centrally symmetric octagons

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 regularn-gon,n 5, is cut intom triangles of equal areas, thenm is a multiple ofn. 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 forn = 6 andn = 8.     

Polytopes with centrally symmetric faces

It is shown that if, for some 2≦j≦d-2, all thej-faces of ad-polytopeP are centrally symmetric, then all the faces ofP of every dimension are centrally symmetric.     

Parallelepipeds and centrally symmetric hexagonal prisms circumscribed about a three-dimensional centrally symmetric convex body

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

Polytopes with centrally symmetric facets

A new and conceptually simpler proof is given of the theorem of A. D. Aleksandrov and G. C. Shephard, that ad-polytope (d≧3), all of whose facets are centrally symmetric, is itself centrally symmetric.     

Finite coverings by translates of centrally symmetric convex domains

Bambah and Rogers proved that the area of a convex domain in the plane which can be covered byn translates of a given centrally symmetric convex domainC is at most (n–1)h(C)+a(C), whereh(C) denotes the area of the largest hexagon contained inC anda(C) stands for the area ofC. An improvement with a term of magnitude n is given here. Our estimate implies that ifC is not a parallelogram, then any covering of any convex domain by at least 26 translates ofC is less economic than the thinnest covering of the whole plane by translates ofC.     

Plane sections of centrally symmetric convex bodies

The note contains an example of three plane convex centrally symmetric figuresP ₁,P ₂,P ₃ such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent toP ₁,P ₂,P ₃ respectively.     

On toric h-vectors of centrally symmetric polytopes

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     

Homothetic packings of centrally symmetric convex bodies

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...     

Completely classifying all vertex-transitive and edge-transitive polyhedra

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.     

Generating the centrally symmetric 3-polyhedral graphs

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.     

Some problems concerning centrally symmetric convex bodies

The conjecture that among convex bodies Q in Rⁿ, 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.     

Equivariant embeddings of low dimensional symmetric planes

We determine the class of all locally compact stable planesM of positive dimensiond4 which admit a reflection at each point of some open setU 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 caseU=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.     

On symmetric incidence matrices of projective planes

We investigate the incidence matrix of a finite plane of ordern which admits a (C, L)-transitivityG. The elation groupG affords a generalized Hadamard matrixH=(h _ij ) of ordern and an incidence matrix for the plane is completely determined by the matrixR(H)=(R(h _ij )), whereR(g) denotes the regular permutation matrix forgG. We prove that in the caseR(H) is symmetric thatG is an elementary abelian 2-group or elseG is a nonabelian group andn is a square. Results are obtained in the abelian case linking the roots of the incidence matrixR(H) to the roots of the complex matrix (H), a nontrivial character ofG.     

A Blichfeldt-type inequality for centrally symmetric convex bodies

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.