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

Yasunori Kimura,Shuta Sudo,Christopher Goodrich

doi:10.1155/2022/9534978

Abstract

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.

Highlights

There are a lot of works dealing with the relation between convergence of a sequence of sets and convergence of a sequence of projections corresponding to it
Let fCng ⊂ 2E be a sequence of nonempty closed convex sets and C0 ⊂ E a nonempty closed convex subset such that fCng converges to C0 in the sense of Mosco
Let fCng ⊂ 2X be a sequence of nonempty closed convex sets and C0 ⊂ X a nonempty closed convex subset such that fCng converges to C0 in the sense of Δ -Mosco

Summary

Introduction

There are a lot of works dealing with the relation between convergence of a sequence of sets and convergence of a sequence of projections corresponding to it. The following theorem on a reflexive and strictly convex real Banach space is one of the important results. Let E be a strictly convex and reflexive real Banach space satisfying the Kadec-Klee property. We know that a Hadamard space is another generalization of Hilbert spaces It is defined as a complete metric space having a particular convexity structure and it has various useful properties that Hilbert spaces have. Kimura and Satô [3] introduced Δκ-Mosco convergence in complete CAT(κ) spaces and obtain the following result: Theorem 3 (Kimura and Satô [3]). We introduce a new concept of the setconvergence and we obtain a similar result as above under the assumptions that both uniformly convex real Banach spaces and complete CAT(0) spaces have

Preliminaries

Uniform Convexity of a Complete Geodesic Space

Δ-Convergence

Convergence of a Sequence of Sets

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