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

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 of high rank for the symmetric groups

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1.     

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.     

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...     

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     

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.     

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.     

Intersecting all edges of centrally symmetric polyhedra by planes

A1 least three planes are needed in order to cut all edges of a 3-dimensional centrally symmetric convex polytope by planes which miss all vertices of the polytope. In contrast, there exist centrally symmetric convex tessellations of the 2-sphere for which two planes are sufficient.     

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.     

The domino inequalities: facets for the symmetric traveling salesman polytope

Adam Letchford defines in [4] the Domino Parity inequalities for the Symmetric Traveling Salesman Polytope (STSP) and gives a polynomial algorithm for the separation of such constraints when the support graph is planar, generalizing a result of Fleischer and Tardos [2] for maximally violated comb inequalities. Naddef in [5] gives a set of necessary conditions for such inequalities to be facet defining for the STSP. These conditions lead to the Domino inequalities and it is shown in [5] that one does not lose any facet inducing inequality restricting the Domino Parity inequalities to Domino inequalities, except maybe for some very particular case. We prove here that all the domino inequalities are facet inducing for the STSP, settling the conjecture given in [5]. As a by product we will also have a complete proof that the comb inequalities are facet inducing. Mathematics Subject Classification (2000):Main 90C57, secondary 90C27     

Lattice Polytopes with Distinct Pair-Sums

Let P be a lattice polytope in R ⁿ , and let P \cap Z ⁿ = {v ₁ ,\ldots,v _N } . If the N + N\choose 2 points 2v ₁ ,\ldots, 2v _N ;v ₁ +v ₂ ,\ldots, v _N-1 + v _N are distinct, we say that P is a ``distinct pair-sum' or ``dps' polytope. We show that if P is a dps polytope in R ⁿ , then N≤ 2 ⁿ , and, for every n , we construct dps polytopes in R ⁿ which contain 2 ⁿ lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials. Received November 10, 2000, and in revised form June 28, 2001. Online publication November 2, 2001.