Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

Abstract and Applied Analysis ◽

10.1155/2022/9534978 ◽

2022 ◽

Vol 2022 ◽

pp. 1-11

Author(s):

Yasunori Kimura ◽

Shuta Sudo

Keyword(s):

Uniform Convexity ◽

Set Convergence ◽

Geodesic Space ◽

Metric Projections ◽

Geodesic Spaces

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

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Convergence of a Sequence of Sets in a Hadamard Space and the Shrinking Projection Method for a Real Hilbert Ball

Abstract and Applied Analysis ◽

10.1155/2010/582475 ◽

2010 ◽

Vol 2010 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

Yasunori Kimura

Keyword(s):

Projection Method ◽

Convergence Theorem ◽

Iterative Scheme ◽

Equivalent Condition ◽

Set Convergence ◽

Hadamard Space ◽

Shrinking Projection Method ◽

Metric Projections ◽

Hilbert Ball

We propose a new concept of set convergence in a Hadamard space and obtain its equivalent condition by using the notion of metric projections. Applying this result, we also prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.

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Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

Axioms ◽

10.3390/axioms10010015 ◽

2021 ◽

Vol 10 (1) ◽

pp. 15

Author(s):

Kengo Kasahara ◽

Yasunori Kimura

Keyword(s):

Finite Number ◽

Convex Functions ◽

Lower Semicontinuous ◽

Resolvent Operators ◽

Geodesic Space ◽

Iterative Schemes ◽

Geodesic Spaces

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.

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Essential circles and Gromov–Hausdorff convergence of covers

Journal of Topology and Analysis ◽

10.1142/s1793525316500047 ◽

2016 ◽

Vol 08 (01) ◽

pp. 89-115

Author(s):

Conrad Plaut ◽

Jay Wilkins

Keyword(s):

Riemannian Manifolds ◽

Minimal Cardinality ◽

Geodesic Space ◽

Limiting Behavior ◽

Hausdorff Convergence ◽

Generating Sets ◽

Finite Set ◽

Geodesic Spaces ◽

Simply Connected ◽

Gromov Hausdorff Convergence

The [Formula: see text]-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of covering maps of compact geodesic spaces called “circle covers” that are “closed” with respect to Gromov–Hausdorff convergence and include [Formula: see text]-covers. In fact, we use circle covers to completely understand the limiting behavior of [Formula: see text]-covers. The proofs use the descrete homotopy methods developed by Berestovskii, Plaut, and Wilkins, and in fact we show that when [Formula: see text], the Sormani–Wei [Formula: see text]-cover is isometric to the Berestovskii–Plaut–Wilkins [Formula: see text]-cover. Of possible independent interest, our arguments involve showing that “almost isometries” between compact geodesic spaces result in explicitly controlled quasi-isometries between their [Formula: see text]-covers. Finally, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.

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On Approximation Methods in Some Geodesic Spaces without the Nice Projection Property

10.20944/preprints202003.0281.v1 ◽

2020 ◽

Author(s):

Pongsakorn Yotkaew

Keyword(s):

Nonexpansive Mappings ◽

Approximation Methods ◽

Geodesic Space ◽

Projection Property ◽

Geodesic Spaces

The purpose of this paper is to prove strong convergent theorems for Browder's type iterations and Halpern's type iterations of a family of nonexpansive mappings in a complete geodesic space with curvature bounded above by a positive number. Moudafi's viscosity type methods are also discussed without the nice projection property.

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On Random Uniform Convexity

Acta Analysis Functionalis Applicata ◽

10.3724/sp.j.1160.2012.00023 ◽

2012 ◽

Vol 14 (1) ◽

pp. 23

Author(s):

Xiaolin ZENG

Keyword(s):

Uniform Convexity

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Some Continuation results in Uniquely Geodesic Spaces

Journal of Physics Conference Series ◽

10.1088/1742-6596/1850/1/012046 ◽

2021 ◽

Vol 1850 (1) ◽

pp. 012046

Author(s):

V.V. Sreya ◽

P. Shaini

Keyword(s):

Geodesic Spaces

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Weighted higher order exponential type inequalities in metric spaces and applications

Georgian Mathematical Journal ◽

10.1515/gmj-2020-2059 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Huiju Wang ◽

Pengcheng Niu

Keyword(s):

Homogeneous Space ◽

Lie Groups ◽

Sobolev Spaces ◽

Exponential Type ◽

Metric Spaces ◽

Type Inequality ◽

Higher Order ◽

Geodesic Space ◽

Stratified Lie Groups ◽

Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

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