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Extreme points and geometric aspects of compact convex sets in asymmetric normed spaces  [PDF]
Natalia Jonard-Pérez,Enrique A. Sánchez-Pérez
Mathematics , 2014,
Abstract: Inspired in the theorem of Krein-Milamn, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all compact convex subsets of a finite dimensional asymmetric normed space.
Geometric properties of Banach spaces and the existence of nearest and farthest points  [PDF]
Stefan Cobza
Abstract and Applied Analysis , 2005, DOI: 10.1155/aaa.2005.259
Abstract: The aim of this paper is to present some generic existence resultsfor nearest and farthest points in connection with some geometricproperties of Banach spaces.
Entropy rigidity of symmetric spaces without focal points  [PDF]
Fran?ois Ledrappier,Lin Shu
Mathematics , 2012,
Abstract: We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.
Geometric Knot Spaces and Polygonal Isotopy  [PDF]
Jorge Alberto Calvo
Mathematics , 1999,
Abstract: The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains only three (when n = 6) or four (when n = 7) topological knot types. Therefore ``geometric knot equivalence'' is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n \ge 8 is also discussed.
Uniform Theory of Geometric Spaces  [PDF]
Alexander Popa
Mathematics , 2010,
Abstract: Isaak Moiseevich Yaglom deduced complete classification of geometric spaces. In this work, supposed to your attention, author formalizes Yaglom's approach and constructs uniform theory of geometric spaces on analytic level. Among its advantages there are its universality and the fact it is easy to use. It isn't limited to specific dimension. The theory becomes the background of the GeomSpace project (http://sourceforge.net/projects/geomspace/).
Uniform Model of Geometric Spaces  [PDF]
Alexandru Popa
Mathematics , 2010,
Abstract: Author developed a uniform model for different spaces where distance and angle measure kinds are parameters. This model is calculus centric, but can also be used in theoretical research. It is useful in the following domains: deduction of uniform equations among geometric spaces, uniform model applied to any space, which provides an easy way to calculate distances, plane and dihedral angles of any dimension, areas and volumes as well as parallel (where applied) and orthogonal property detection, study of not yet described spaces and more.
Conjugate Points in Length Spaces  [PDF]
Krishnan Shankar,Christina Sormani
Mathematics , 2007,
Abstract: In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\rm CBA}(\kappa)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $\pi$ apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems.
Geometric Computations on Indecisive and Uncertain Points  [PDF]
Allan Jorgensen,Maarten L?ffler,Jeff M. Phillips
Computer Science , 2012,
Abstract: We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a finite number of locations, the points are called indecisive points. In particular, we focus on geometric shape-fitting problems and on building compact distributions to describe how the solutions to these problems vary with respect to the uncertainty in the points. Our main results are: (1) a simple and efficient randomized approximation algorithm for calculating the distribution of any statistic on uncertain data sets; (2) a polynomial, deterministic and exact algorithm for computing the distribution of answers for any LP-type problem on an indecisive point set; and (3) the development of shape inclusion probability (SIP) functions which captures the ambient distribution of shapes fit to uncertain or indecisive point sets and are admissible to the two algorithmic constructions.
Geometric formality of homogeneous spaces and of biquotients  [PDF]
D. Kotschick,S. Terzic
Mathematics , 2009, DOI: 10.2140/pjm.2011.249.157
Abstract: We provide examples of homogeneous spaces which are neither symmetric spaces nor real cohomology spheres, yet have the property that every invariant metric is geometrically formal. We also extend the known obstructions to geometric formality to some new classes of homogeneous spaces and of biquotients, and to certain sphere bundles.
The geometric Hopf invariant and double points  [PDF]
Michael Crabb,Andrew Ranicki
Mathematics , 2010,
Abstract: The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set of f. We interpret the Smale-Hirsch-Haefliger regular homotopy classification of immersions f in the metastable dimension range 3m<2n-1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.
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