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  • 1990-1994  (7)
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  • 1
    Electronic Resource
    Electronic Resource
    Realizations of regular apeirotopes (1994)
    Springer
    Aequationes mathematicae 47 (1994), S. 223-239 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01832961
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Monotone translation invariant valuations on convex bodies (1990)
    Springer
    Archiv der Mathematik 55 (1990), S. 595-598 
    Details
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01191696
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On simple polytopes (1993)
    Springer
    Inventiones mathematicae 113 (1993), S. 419-444 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01244313
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Inequalities between intrinsic volumes (1991)
    Springer
    Monatshefte für Mathematik 111 (1991), S. 47-53 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01299276
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  • 5
    Electronic Resource
    Electronic Resource
    The regular polyhedra of type {p, 3} with 2p vertices (1992)
    Springer
    Geometriae dedicata 43 (1992), S. 285-289 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF00151518
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Duality, sections and projections of certain Euclidean tilings (1994)
    Springer
    Geometriae dedicata 49 (1994), S. 183-202 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01610620
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    Paper (German National Licenses)
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  • 7
    Book
    Book
    Polytopes: abstract, convex and computational. Proceedings of the NATO Advanced Study Institute, Scarborough, Ontario, Canada, August 20 - September 3, 1993; 440 (1994)
    Dordrecht, Netherlands :Kluwer,
    Details
    Title: Polytopes: abstract, convex and computational. Proceedings of the NATO Advanced Study Institute, Scarborough, Ontario, Canada, August 20 - September 3, 1993; 440
    Contributer: Bisztriczky, Tibor , McMullen, Peter , Weiss, A. Ivic , Schneider, Rolf
    Publisher: Dordrecht, Netherlands :Kluwer,
    Year of publication: 1994
    Pages: 507 S.
    Series Statement: NATO ASI series C: Mathematical and physical sciences 440
    Type of Medium: Book
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