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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Aequationes mathematicae 36 (1988), S. 320-320 
    ISSN: 1420-8903
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Aequationes mathematicae 37 (1989), S. 38-56 
    ISSN: 1420-8903
    Keywords: Primary 51M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let ℐ be a finite regular incidence-polytope. A realization of ℐ is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group Γ(ℐ) of ℐ induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of ℐ forms, in a natural way, a closed convex cone, which is also denoted by ℐ The dimensionr of ℐ is the number of equivalence classes under Γ(ℐ) of diagonals of ℐ, and is also the number of unions of double cosets Γ*σΓ* ∪ Γ*σ−1Γ* (σ ∉ Γ*), where Γ* is the subgroup of Γ(ℐ) which fixes some given vertex of ℐ. The fine structure of ℐ corresponds to the irreducible orthogonal representations of Γ(ℐ). IfG is such a representation, let its degree bed G , and let the subgroup ofG corresponding to Γ* have a fixed space of dimensionw G . Then the relations $$\begin{array}{l} \Sigma _G w_G d_G = \upsilon - 1, \\ \Sigma _G {\textstyle{1 \over 2}}w_G (w_G + 1) = r, \\ \Sigma _G w_G ^2 = \bar w \\ \end{array}$$ hold, where ℐ hasv vertices, and $$\bar w$$ is the number of double cosets Γ*σΓ* (σ ∉ Γ*). The second relation corresponds to the fact that the realizations associated with a given irreducible representationG form a cone of dimension 1/2w G (w G + 1), which forw G ⩾ 2 has as base the convex hull of a projective space of dimensionw G − 1 embedded in an ellipsoid of dimension 1/2w G (w G + 1) − 2. Comparison of the second and third relations leads to a curious connexion between the cone ℐ and the group Γ(ℐ), namely, that the following conditions are equivalent: (1)r = $$\bar w$$ , (2) ℐ is polyhedral, (3)w G ⩽ 1 for all irreducible orthogonal representationsG of Γ(ℐ), (4) σ-1 ∈ Γ*σΓ* for each σ ∈ Γ(ℐ). The realization cones ℐ are described for various regular polytopes.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Archiv der Mathematik 41 (1983), S. 555-564 
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Aequationes mathematicae 47 (1994), S. 223-239 
    ISSN: 1420-8903
    Keywords: 51M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In an earlier paper, a theory of realizations of (finite) regular polytopes in euclidean spaces was developed. Here, the analogous problem of realizing regular apeirotopes (infinite polytopes) is investigated. While no complete theory is expounded, several basic results are established. Among these is the curious fact that, if a regular apeirotope has a discrete realization, then it has one with no translations in its symmetry group.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Archiv der Mathematik 55 (1990), S. 595-598 
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Inventiones mathematicae 113 (1993), S. 419-444 
    ISSN: 1432-1297
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetP be a simpled-polytope ind-dimensional euclidean space $$\mathbb{E}^d $$ , and let Π(P) be the subalgebra of the polytope algebra Π generated by the classes of summands ofP. It is shown that the dimensions of the weight spacesΞ r(P) of Π(P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality betweenΞ r (P) andΞ d-r (P). Moreover, Π(P) admits a Lefschetz decomposition under multiplication by the element ofΞ 1(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between Π(P) and the face ring of the dual simplicial polytopeP * enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP * is also established.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Monatshefte für Mathematik 104 (1987), S. 265-272 
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The centrally symmetric convex polytopes whose images under orthogonal projection on to any pair of orthogonal complementary subspaces ofE d have numerically equal volumes are shown hare to be certain cartesian products of polygons and line segments. Ford≥3, the general projection property in fact follows from that for pairs of hyperplanes and lines. A conjecture is made about the problem in the non-centrally symmetric case.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Monatshefte für Mathematik 111 (1991), S. 47-53 
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Removing the dependence on dimension of the inequalities between quermassintegrals resulting from the Aleksandrov-Fenchel inequalities leads to universal quadratic inequalities between intrinsic volumes, and to an inequality for the Wills functional. The inequalities correspond to equations which hold in the polytope algebra.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Geometriae dedicata 43 (1992), S. 285-289 
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract G(p, d) is a cubic (3-valent) graph consisting of a p-gon and a (p/d)-gon (a starpolygon) with corresponding vertices joined (the notation admits anomalous cases, when d=1 or (d, p)〉1), and with a high degree of symmetry. It is shown here that the seven possible graphs G(p, d) are just the edge-graphs of the regular polyhedra of type {p, 3} with 2p vertices, and therefore 3p edges, 6 faces, and symmetry group of order 12p.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Geometriae dedicata 49 (1994), S. 183-202 
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Pegged tilings localize the defining property of or Laguerre tilings, and, like them, admit a natural duality (corresponding to the Delaunay tilings of tilings). It can thus be shown that the projection method, which is generally used to construct quasi-periodic tilings related to tilings of higher dimensional lattices, applies to this wider class of tilings. Of further importance is that pegged tilings are just those which can be lifted to the graphs of convex functions with a certain strong locally polyhedrality property. The context of convex functions also provides a direct way of viewing the projection method, and leads to alternative pictures of special cases such as various grid methods.
    Type of Medium: Electronic Resource
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