不是最远点的Chebyshev中心

Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

In this paper, we prove that if $E$ is a uniquely remotal subset of a real normed linear space $X$ such that $E$ has a Chebyshev center $c \in X$ and the farthest point map $F:X\rightarrow E$ restricted to $[c,F(c)]$ is partially statistically continuous at $c$, then $E$ is a singleton. We obtain a necessary condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set $M$ having a Chebyshev center $c$ such that the farthest point map $F:\mathbb{R}\rightarrow M$ is not continuous at $c$ but is partially statisticall…

Restricted Chebyshev centers in $L_1$-predual spaces

In this paper, we provide a necessary and sufficient condition for the existence of a restricted Chebyshev center of a compact subset of an $L_{1}$-predual space in a closed convex subset of the $L_{1}$-predual space. We also provide a geometrical characterization of an $L_{1}$-predual space in terms of the restricted Chebyshev radius in the following manner. A real Banach space $X$ is an $L_{1}$-predual space if and only if for each non-empty finite subset $F$ of $X$ and closed convex subset $V$ of $X$, $rad_{V}(F) = rad_{X}(F) + d(V, cent_{X}(F))$, where we denote $rad_{X}(F)$, $rad_{V}(F)$,…

关于Banach空间中的Chebyshev中心

On Chebyshev centers in Banach spaces

本文研究了Banach空间中Chebyshev中心的存在性，并证明了(GC)类不是3空间性质。

On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums

For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…

Some stability properties of restricted Chebyshev centers in Banach spaces

In this paper, we discuss the stability of (restricted) Chebyshev centers in few function spaces. For an extremally disconnected compact Hausdorff space $K$ and a finite dimensional Banach space $X$, we prove the existence of Chebyshev centers of closed bounded subsets of the space of $X$-valued continuous functions on $K$, $C(K,X)$ and that of compact subsets of $M$-ideals in $C(K,X)$. It is also proved that the existence of restricted Chebyshev centers is stable in the space of vector-valued bounded functions on an arbitrary set. Furthermore, the stability of continuity properties of the Che…

论Banach空间中的性质-$(R_1)$与相对Chebyshev中心-II

On property-$\bm{(R_1)}$ and relative Chebyshev centers in Banach spaces-II

该研究延续探讨Banach空间中的（强）性质-(R₁)，通过分析Lindenstrauss空间等场景，证明相关紧子集的相对Chebyshev中心非空且具Lipschitz连续性，拓展了经典结论，并推导其在连续函数空间等的稳定性。