论文标题
在物业上 - $ \ bm {（r_1）} $和相对的chebyshev中心在Banach Spaces-II中
On property-$\bm{(R_1)}$ and relative Chebyshev centers in Banach spaces-II
论文作者
论文摘要
我们继续研究（强）属性 - $（R_1）$在Banach空间中。正如Pai \＆Nowroji在[{\ it of sets的受限中心}中所讨论的。 Theory, {\bf 66}(2), 170--189 (1991)], this study corresponds to a triplet $(X,V,\mathcal{F})$, where $X$ is a Banach space, $V$ is a closed convex set, and $\mathcal{F}$ is a subfamily of closed, bounded subsets of $X$.可以观察到，如果$ x $是lindenstrauss空间，则$（x，b_x，\ mathcal {k}（x））$具有强属性 - $（r_1）$，其中$ \ mathcal {k}（x）$代表$ x $的紧凑型子集。可以确定，对于任何$ f \ in \ Mathcal {k}（x）$，$ \ textrm {cent} _ {b_x}}（f）\ neq \ emptyset $。这扩展了一个众所周知的事实，即Lindenstrauss Space的紧凑子集$ X $承认了$ x $的非空置Chebyshev中心。我们扩展了$ \ textrm {cent} _ {b_x} $是lipschitz在$ \ mathcal {k}（x）$中连续的观察结果，如果$ x $是lindenstrauss的空间。如果$ y $是Banach Space $ x $和$ \ MATHCAL {F} $的子空间，则代表$ b_x $的所有有限子集的集合，那么我们观察到，$ b_y $同时在$ f $ f $ f \ y y y y y y y y y y y y y y y y y y in $ x $的情况下同时表现出强烈的近端（$（p_1）$）的条件。 \ Mathcal {f}（x））$满足强属性 - $（r_1）$，其中$ \ Mathcal {f}（x）$代表$ x $的所有有限子集的集合。 It is demonstrated that if $P$ is a bi-contractive projection in $\ell_\infty$, then $(\ell_\infty, Range (P), \mathcal{K}(\ell_\infty))$ exhibits the strong property-$(R_1)$, where $\mathcal{K}(\ell_\infty)$ represents the set of all compact subsets of $ \ ell_ \ infty $。此外，这些性质的稳定性结果是在连续函数空间中得出的，然后在Banach空间中研究了这些属性的稳定性结果。
We continue to study (strong) property-$(R_1)$ in Banach spaces. As discussed by Pai \& Nowroji in [{\it On restricted centers of sets}, J. Approx. Theory, {\bf 66}(2), 170--189 (1991)], this study corresponds to a triplet $(X,V,\mathcal{F})$, where $X$ is a Banach space, $V$ is a closed convex set, and $\mathcal{F}$ is a subfamily of closed, bounded subsets of $X$. It is observed that if $X$ is a Lindenstrauss space then $(X,B_X,\mathcal{K}(X))$ has strong property-$(R_1)$, where $\mathcal{K}(X)$ represents the compact subsets of $X$. It is established that for any $F\in\mathcal{K}(X)$, $\textrm{Cent}_{B_X}(F)\neq\emptyset$. This extends the well-known fact that a compact subset of a Lindenstrauss space $X$ admits a nonempty Chebyshev center in $X$. We extend our observation that $\textrm{Cent}_{B_X}$ is Lipschitz continuous in $\mathcal{K}(X)$ if $X$ is a Lindenstrauss space. If $Y$ is a subspace of a Banach space $X$ and $\mathcal{F}$ represents the set of all finite subsets of $B_X$ then we observe that $B_Y$ exhibits the condition for simultaneously strongly proximinal (viz. property-$(P_1)$) in $X$ for $F\in\mathcal{F}$ if $(X, Y, \mathcal{F}(X))$ satisfies strong property-$(R_1)$, where $\mathcal{F}(X)$ represents the set of all finite subsets of $X$. It is demonstrated that if $P$ is a bi-contractive projection in $\ell_\infty$, then $(\ell_\infty, Range (P), \mathcal{K}(\ell_\infty))$ exhibits the strong property-$(R_1)$, where $\mathcal{K}(\ell_\infty)$ represents the set of all compact subsets of $\ell_\infty$. Furthermore, stability results for these properties are derived in continuous function spaces, which are then studied for various sums in Banach spaces.
