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51. CMB 2002 (vol 45 pp. 537)

Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei
Polytopal Realizations of Generalized Associahedra
No abstract.

Categories:05E15, 20F55, 52C07

52. CMB 2002 (vol 45 pp. 232)

Ji, Min; Shen, Zhongmin
On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35

53. CMB 2002 (vol 45 pp. 123)

Moody, Robert V.
Uniform Distribution in Model Sets
We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the `physical') space and its internal space. We prove, assuming only that the window defining the model set is measurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

Categories:52C23, 11K70, 28D05, 37A30

54. CMB 2000 (vol 43 pp. 427)

Ivey, Thomas A.
Helices, Hasimoto Surfaces and Bäcklund Transformations
Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in $\R^3$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the B\"acklund transformation for constant torsion curves in $\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in $H^3$ or $S^3$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

Keywords:surfaces, filament flow, Bäcklund transformations
Categories:53A05, 58F37, 52C42, 58A15

55. CMB 2000 (vol 43 pp. 368)

Litvak, A. E.
Kahane-Khinchin's Inequality for Quasi-Norms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the ``tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman).

Categories:46B09, 52A30, 60B11

56. CMB 1999 (vol 42 pp. 380)

Nurmela, Kari J.; Östergård, Patric R. J.; aus dem Spring, Rainer
Asymptotic Behavior of Optimal Circle Packings in a Square
A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

Keywords:asymptotic bound, circle packing
Category:52C15

57. CMB 1999 (vol 42 pp. 237)

Thompson, A. C.
On Benson's Definition of Area in Minkowski Space
Let $(X, \norm)$ be a Minkowski space (finite dimensional Banach space) with unit ball $B$. Various definitions of surface area are possible in $X$. Here we explore the one given by Benson \cite{ben1}, \cite{ben2}. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

Categories:52A21, 52A38
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