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26. CJM 2005 (vol 57 pp. 844)

Williams, Gordon
 Petrie Schemes Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Gr\"unbaum--Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter groupCategories:52B15, 52B05

27. CJM 2004 (vol 56 pp. 529)

Martínez-Finkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.
 Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$ We consider the $s$-energy $$E(\ZZ_n;s)=\sum_{i \neq j} K(\|z_{i,n}-z_{j,n}\|;s)$$ for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets $\Ga$ in $\R^d$ having finite one-dimensional Hausdorff measure, where $$K(t;s)= \begin{cases} t^{-s} ,& \mbox{if } s>0, \\ -\ln t, & \mbox{if } s=0, \end{cases}$$ is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\geq 1$, the minimizing nodes for a rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with respect to arclength as $n\to\infty$. Keywords:Riesz energy, Minimal discrete energy,, Rectifiable curves, Best-packing on curvesCategories:52A40, 31C20

28. CJM 2004 (vol 56 pp. 472)

 Infinite-Dimensional Polyhedrality This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a \emph{polytope} if each of its finite-dimensional affine sections is a (standard) polytope. We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open). Categories:46B20, 46B03, 46B04, 52B99

29. CJM 2001 (vol 53 pp. 1121)

 Monotone Paths on Zonotopes and Oriented Matroids Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\lceil 2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d \ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included. Categories:52C35, 52B12, 52C40, 20F55

30. CJM 2001 (vol 53 pp. 470)

Bauschke, Heinz H.; Güler, Osman; Lewis, Adrian S.; Sendov, Hristo S.
 Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, G{\aa}rding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize G{\aa}rding's result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones. Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric functionCategories:90C25, 15A45, 52A41

31. CJM 1999 (vol 51 pp. 1258)

Baake, Michael; Moody, Robert V.
 Similarity Submodules and Root Systems in Four Dimensions Lattices and $\ZZ$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties. Categories:11S45, 11H05, 52C07

32. CJM 1999 (vol 51 pp. 1300)

Conway, J. H.; Rains, E. M.; Sloane, N. J. A.
 On the Existence of Similar Sublattices Partial answers are given to two questions. When does a lattice $\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is geometrically similar to $\Lambda$? When is the sublattice clean'', in the sense that the boundaries of the Voronoi cells for $\Lambda'$ do not intersect $\Lambda$? Category:52C07

33. CJM 1999 (vol 51 pp. 449)

Bahn, Hyoungsick; Ehrlich, Paul
 A Brunn-Minkowski Type Theorem on the Minkowski Spacetime In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also present an isoperimetric inequality in the Minkowski spacetime $\mathbb{L}^{n+1}$ as a consequence of this Brunn-Minkowski type theorem. Keywords:Minkowski spacetime, Brunn-Minkowski inequality, isoperimetric inequalityCategories:53B30, 52A40, 52A38

34. CJM 1999 (vol 51 pp. 225)

Betke, U.; Böröczky, K.
 Asymptotic Formulae for the Lattice Point Enumerator Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon (d)})$ for some positive $\varepsilon(d)$. Here we give for general convex bodies the weaker estimate $\left| G(\lambda M) -V(\lambda M) \right | \le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1})$ where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term $S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$. Further we deal with families $\{P_\lambda\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have $\left| G(P_\lambda)-V(P_\lambda) \right| \le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr)$ where the convex body $K$ satisfies some simple condition, $V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the surface area of $P_\lambda$. Categories:11P21, 52C07

35. CJM 1998 (vol 50 pp. 426)

McMullen, Peter
 The groups of the regular star-polytopes No abstract. Categories:51M20, 52C99

36. CJM 1998 (vol 50 pp. 16)

Böröczky, Károly; Schnell, Uwe
 Asymptotic shape of finite packings Let $K$ be a convex body in $\ed$ and denote by $\cn$ the set of centroids of $n$ non-overlapping translates of $K$. For $\varrho>0$, assume that the parallel body $\cocn+\varrho K$ of $\cocn$ has minimal volume. The notion of parametric density (see~\cite{Wil93}) provides a bridge between finite and infinite packings (see~\cite{BHW94} or~\cite{Hen}). It is known that there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that $\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}). We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that if $\varrho>\varrho_c$ and $n$ is large then the shape of $\cocn$ can not be too far from the shape of $K$. For $d=2$, we verify that $\varrho_s=\varrho_c$. For $d\geq 3$, we present the first example of a convex body with known $\varrho_s$ and $\varrho_c$; namely, we have $\varrho_s=\varrho_c=1$ for the parallelotope. Categories:52C17, 05B40

37. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
 Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature. Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalitiesCategories:51M10, 51M25, 52A40, 53C20
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