October 02, 2025
We study the stability of various restricted center properties and certain continuity properties of the restricted center map. We observe that restricted center property, property-\((P_1)\) and semi-continuity properties of the restricted center map are preserved under \(\ell_p\)-direct sum \((1 \leq p < \infty).\) It is shown that property-\((P_2)\) is stable under finite \(\ell_p\)-direct sum, but not under infinite \(\ell_p\)-direct sum. Additionally, we introduce a notion called property-\((lP_2)\) as a sufficient condition for the continuity of the restricted center map. Further, the stability of property-\((lP_2)\) is established.
Let \(X\) be a real Banach space. For any non-empty subset \(M\) of \(X,\) we denote the set of all non-empty closed subsets of \(M\), non-empty closed bounded subsets of \(M\), non-empty closed convex subsets of \(M\) and non-empty compact subsets of \(M\) respectively by \(CL(M)\), \(CB(M)\), \(CC(M)\) and \(K(M).\) The closed unit ball and unit sphere of \(X\) are denoted by \(B_X\) and \(S_X,\) respectively.
For any \(x\in X\) and \(F \in CB(X),\) we define \(r(x, F)=\sup\{\|x-a\| : a \in F\}.\) The restricted radius of \(F\) relative to \(V\in CL(X),\) denoted by \(rad_V(F),\) is defined as \(rad_V(F)=\inf\{r(v, F): v \in V \}.\) The set \(Z_V(F)\) of all points from \(V\) at which the infimum in \(rad_V(F)\) attain is called restricted centers of \(F\) relative to \(V,\) i.e., \(Z_V(F)=\{v \in V : r(v, F)=rad_V(F)\}.\)
We say that the pair \((V, F)\) has restricted center property (in short, r.c.p.) if \(Z_V(F)\) is non-empty. Further, we say that the pair \((V, \mathcal{F})\) has r.c.p. if \((V,G)\) has r.c.p. for every \(G \in \mathcal{F}.\) A sequence \((v_n)\) in \(V\) is said to be a minimizing sequence for \(F\) if \(r(v_n, F) \to rad_V(F).\) For any \(\mathcal{F} \subseteq CB(X),\) the map \(Z_V\) from \((\mathcal{F}, \mathcal{H})\) to \((CB(V), \mathcal{H})\), defined as \(F \mapsto Z_V(F)\), is said to be the restricted center map. Here, \(\mathcal{H}\) is the Hausdorff metric on \(CB(X)\) given by \(\mathcal{H}(A,B)= \inf\{r>0: A \subseteq B+ rB_X \;and \;B \subseteq A+ rB_X \}\) for any \(A, B \in CB(X)\).
To study the continuity properties of the restricted center map, Mach [1] introduced two notions called property-\((P_1)\) and property-\((P_2).\) To see some geometric aspects of these notions recently Lalithambigai et al. [2] equivalently reformulated these notions as follows. For any \(\delta \geq 0,\) we denote \(Z_V(F, \delta)=\{v \in V: r(v,F)\leq rad_V(F)+ \delta\}.\) Clearly \(Z_V(F,0)= Z_V(F).\)
Definition 1. Let \(V \in CL(X)\), \(F\in CB(X)\) and \(\mathcal{F} \subseteq CB(X).\)
We say that the pair \((V, F)\) has property-\((P_1)\) if \((V, F)\) has r.c.p. and for every \(\epsilon>0\) there exists \(\delta>0\) such that \(Z_V(F, \delta) \subseteq Z_V(F)+\epsilon B_X.\) We say that the pair \((V, \mathcal{F})\) has property-\((P_1)\) if \((V, G)\) has property-\((P_1)\) for every \(G \in \mathcal{F}.\)
We say that the pair \((V, \mathcal{F})\) has property-\((P_2)\) if \((V, \mathcal{F})\) has r.c.p. and for every \(\epsilon>0\) there exists \(\delta>0\) such that \(Z_V(G, \delta) \subseteq Z_V(G)+\epsilon B_X\) for every \(G \in \mathcal{F}.\)
In [1], Mach observed that if \((V,\mathcal{F})\) has property-\((P_1)\), then the restricted center map \(Z_V\) is upper Hausdorff semi-continuous on \((\mathcal{F},\mathcal{H}).\) Also, [1], it is shown that the restricted center map \(Z_V\) is Hausdorff continuous on \((\mathcal{F}, \mathcal{H})\), whenever \((V,\mathcal{F})\) has property-\((P_2).\) However, the later result was recently improved, by Lalithambigai et al. [2], by showing that property-\((P_2)\) leads to the uniform Hausdorff continuity of the restricted center map. It is easy to check that the converses of both results are not necessarily true in general, for instance see [2]. At this stage, it is relevant to investigate the possibility of defining an intermediate notion that serves as a sufficient condition for the continuity of the restricted center map. To address this, in this article, we introduce a notion called property-\((lP_2)\).
The theory of best approximation in Banach spaces provides a straightforward approach to explore these restricted center properties through their counterparts. Let \(V \in CL(X)\). If \(F=\{x\}\) for some \(x \in X\), then \(rad_V(F)\) is the distance from \(x\) to \(V,\) denoted by \(d(x,V)\), and \(Z_V(F)\) is the set of all best approximants to \(x\) from \(V,\) i.e., \(P_V(x)=\{v \in V: \|x-v\|=d(x,V)\}.\) Let \(A\) be any non-empty subset of \(X\) and \(\mathcal{F}\) be the collection of all singleton subsets of \(A.\) We say that the set \(V\) is proximinal (respectively, strongly proximinal, uniformly strongly proximinal) on \(A\) if \((V, \mathcal{F})\) has r.c.p. (respectively, property-\((P_1)\), property-\((P_2)\)). Further, we say that the set \(V\) is approximatively compact on \(A\) if it is strongly proximinal on \(A\) and \(P_V(x)\) is compact for every \(x \in A.\) The map \(P_V\) from \(A\) to \((CB(V), \mathcal{H})\) defined as \(x \mapsto P_V(x)\), is called the metric projection. These proximinality notions are closely associated with some geometric properties and provide various continuity properties of the metric projection. We refer to [3], [4] for more results on these notions.
Let \(1 \leq p < \infty.\) In [5], Lalithambigai and Narayana observed the stability of the strong proximinality under finite \(\ell_p\)-direct sum. Further, in the same article, it is proved that the lower Hausdorff semi-continuity of the metric projection is stable under finite \(\ell_{p}\)-direct sum. The stability of approximative compactness under finite \(\ell_{p}\)-direct sum is proved by Bandyopadhyay et al. [6]. This result was later extended to infinite collection by Luo et al. in [7].
It is interesting to study whether analogous stability results hold for restricted center properties and for the continuity properties of restricted center map. Such a question is not new and has already been raised in the literature (see, [8]). In this paper, we present some affirmative answers to this question. We obtain some stability results of restricted center property, property-\((P_1)\), property-\((P_2)\) and the continuity properties of the restricted center map.
The article is organized as follows. The stability of the restricted center property under \(\ell_p\)-direct sum \((1\leq p <\infty)\) is shown in 2. By obtaining some equivalent formulations of property-\((P_2)\), we demonstrate that the notions quasi uniform rotundity with respect to a subspace coincide with the property-\((P_2)\), which further equivalent to the uniform Hausdorff continuity of the restricted center map.
3 deals with the stability of properties-\((P_1)\) and \((P_2)\) under \(\ell_p\)-direct sum \((1 \leq p < \infty)\). We observe that property-\((P_1)\) is stable under infinite \(\ell_p\)-direct sum, but not property-\((P_2)\). However, we prove that property-\((P_2)\) is stable under finite \(\ell_p\)-direct sum. As a sufficient condition for the continuity of restricted center map, we introduce a notion called property-\((lP_2),\) which is intermediate between properties-\((P_1)\) and \((P_2).\) We notice that in spaces which are uniformly rotund in every direction property-\((P_1)\) and property-\((lP_2)\) coincide for closed convex sets. Also, we prove that in finite dimensional spaces the notions property-\((lP_2)\) and property-\((P_2)\) coincide for subspaces. Additionally, we present the stability of property-\((lP_2)\) by obtaining the stability of continuity properties of the restricted center map.
In this section, we obtain the stability of restricted center property and relate property-\((P_2)\) with some geometric notions. These results are essential to prove our main results in the next section. Throughout the paper, we assume all subspaces to be closed.
Let \(I\) be any index set and \(\{X_i : {i\in I}\}\) be a collection of Banach spaces. For \(1 \leq p < \infty,\) consider \(X=(\oplus_pX_i)_{i \in I}\) and denote \[\mathcal{P}(X)=\{F=\Pi_{i \in I } F_i \in CB(X): F_i \in CB(X_i), i \in I\}.\]
The following restricted radius formula is essential to prove the stability of the restricted center property.
Proposition 1. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}},\) \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\) and \(F=\Pi_{i \in \mathbb{N}}F_i \in \mathcal{P}(X).\) Then for any \(x=(x_i)_{i \in \mathbb{N}} \in X,\) we have
\(r(x,F) = \left(\sum_{i\in\mathbb{N}} r(x_i, F_i)^p\right)^{\frac{1}{p}}\);
\(rad_Y(F) = \left(\sum_{i\in\mathbb{N}} rad_{Y_i}(F_i)^p\right)^{\frac{1}{p}}\).
Proof. \((1)\): Let \(x=(x_i)_{i \in \mathbb{N}} \in X\). For every \(i \in \mathbb{N},\) let \((z_{n,i})_{n \in
\mathbb{N}}\) be a sequence in \(F_i\) such that \(\|x_i-z_{n,i}\| \to r(x_i, F_i).\) For every \(n \in \mathbb{N},\) consider \(z_n=(z_{n,i})_{i \in \mathbb{N}} \in F.\) Let \(m \in \mathbb{N}\). Since for any \(n \in \mathbb{N}\) \[\sum\limits_{i =1}^m
\|x_i-z_{n,i}\|^p \leq \sum\limits_{i \in \mathbb{N}} \|x_i-z_{n,i}\|^p = \|x-z_n\|^p \leq r(x, F)^p,\] we have \(\sum_{i =1}^m r(x_i, F_i)^p \leq r(x, F)^p.\) Thus, \(\sum_{i \in
\mathbb{N}} r(x_i, F_i)^p \leq r(x, F)^p.\) Observe that for any \(z=(z_i)_{i \in \mathbb{N}} \in F\), we have \[\|x-z\|^p = \sum_{i\in\mathbb{N}}\|x_i-z_i\|^p \leq \sum_{i\in\mathbb{N}}
r(x_i, F_i)^p.\] Therefore, \(r(x,F)^p \leq \sum_{i\in\mathbb{N}} r(x_i, F_i)^p.\) Hence the equality holds.
\((2)\): For any \(y=(y_i)_{i \in \mathbb{N}}\in Y,\) by \((1),\) it follows that \[\sum\limits_{i \in \mathbb{N}}
rad_{Y_i}(F_i)^p \leq \sum\limits_{i \in \mathbb{N}} r(y_i, F_i)^p = r(y, F)^p.\] Thus, \(\sum_{i \in \mathbb{N}} rad_{Y_i}(F_i)^p \leq rad_Y(F)^p.\) Now to prove the other inequality, for any \(i \in \mathbb{N},\) choose a sequence \((y_{n,i})_{n \in \mathbb{N}}\) in \(Y_i\) such that \(r(y_{n,i}, F_i) \to
rad_{Y_i}(F_i).\) Let \(m \in \mathbb{N}\). For every \(n \in \mathbb{N},\) define \(z_n = (y_{n,1}, y_{n,2},\ldots,y_{n,m},0,0,\ldots) \in Y.\)
Since, by \((1),\) \[rad_Y(F)^p \leq r(z_n, F)^p = \sum\limits_{i =1}^m r(y_{n,i}, F_i)^p + \sum\limits_{i > m} r(0, F_i)^p\] holds for any \(n \in
\mathbb{N},\) we have \(rad_Y(F)^p \leq \sum_{i =1}^m rad_{Y_i}(F_i)^p + \sum_{i > m} r(0, F_i)^p .\) Thus, \(rad_Y(F)^p \leq \sum_{i \in \mathbb{N}} rad_{Y_i}(F_i)^p.\) Hence the
equality holds. ◻
Theorem 2. Let \(\{X_i : {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\), \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\) and \(F= \Pi_{i \in \mathbb{N}}F_i \in \mathcal{P}(X).\) Then \((Y_i, F_i)\) has r.c.p. for every \(i \in \mathbb{N}\) if and only if \((Y, F)\) has r.c.p. Further, we have \(Z_Y(F)=(\oplus_pZ_{Y_i}(F_i))_{i \in \mathbb{N}}= \Pi_{i \in \mathbb{N}}Z_{Y_i}(F_i).\)
Proof. Let \((Y_i, F_i)\) has r.c.p. for every \(i \in \mathbb{N}.\) Choose \(y_i \in Z_{Y_i}(F_i)\) for every \(i \in \mathbb{N}\) and consider \(y=(y_i)_{i \in \mathbb{N}}.\) First we prove that \(y \in Y.\) For this let \(z=(z_i)_{i\in\mathbb{N}} \in F.\) Since for every \(i \in \mathbb{N},\) we have \(\|y_i-z_i\| \leq r(y_i, F_i)\) which implies \(\| y_i\| \leq rad_{Y_i}(F_i)+\|z_i\|.\) Thus, it follows from 1, \[\left(\sum_{i \in \mathbb{N}} \|y_i\|^p\right)^\frac{1}{p} \leq \left(\sum_{i \in \mathbb{N}} rad_{Y_i}(F_i)^p\right)^\frac{1}{p}+ \left(\sum_{i \in \mathbb{N}}\|z_i\|)^p\right)^\frac{1}{p} = rad_Y(F)+\|z\|\] and hence \(y \in Y.\) Further, by 1, \(y \in Z_Y(F).\) Thus, \((Y,F)\) has r.c.p. From the preceding argument, observe that \(\Pi_{i \in \mathbb{N}}Z_{Y_i}(F_i)=(\oplus_p Z_{Y_i}(F_i))_{i \in \mathbb{N}} \subseteq Z_Y(F).\) To prove the converse, let \((Y, F)\) has r.c.p. and \(w=(w_i)_{i \in \mathbb{N}} \in Z_Y(F).\) We claim that \(w_i \in Z_{Y_i}(F_i)\) for every \(i \in \mathbb{N}.\) Suppose there exists \(j \in \mathbb{N}\) such that \(w_j \notin Z_{Y_j}(F_j).\) Since \(rad_{Y_j}(F_j) < r(w_j, F_j)\) and \(rad_{Y_i}(F_i) \leq r(w_i, F_i)\) for every \(i \in \mathbb{N},\) by 1, we have \(rad_Y(F) < r(w, F).\) This is a contradiction. Therefore, \(w_i \in Z_{Y_i}(F_i)\) for every \(i \in \mathbb{N}.\) Hence, \((Y_i, F_i)\) has r.c.p. for every \(i \in \mathbb{N},\) further it follows that \(Z_Y(F) \subseteq (\oplus_p Z_{Y_i}(F_i))_{i \in \mathbb{N}}.\) ◻
Corollary 1. Let \(\{X_i : {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\), \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\) and \(F= \Pi_{i \in \mathbb{N}}F_i \in \mathcal{P}(X).\) Then \(Z_Y(F)\) is singleton if and only if \(Z_{Y_i}(F_i)\) is singleton for every \(i \in \mathbb{N}.\)
It is easy to verify that for any subspace \(Y\) of \(X\), if \((Y, \mathcal{F})\) has r.c.p. (respectively, property-\((P_1)\)) where \(\mathcal{F}=\{F \in CB(X): rad_Y(F)=1\}\), then \((Y, CB(X))\) has r.c.p. (respectively, property-\((P_1)\)). We need this fact and the following equivalences of property-\((P_2)\) to prove our main results in the next section.
Proposition 3. Let \(Y\) be a subspace of \(X\) with \((Y, CB(X))\) has r.c.p. and \(\alpha>0\). Then the following statements are equivalent.
\((Y, \mathcal{F}_1)\) has property-\((P_2)\), where \(\mathcal{F}_1= \{F \in CB(X): rad_Y(F)= r(0,F)=1\}.\)
\((Y, \mathcal{F}_2)\) has property-\((P_2)\), where \(\mathcal{F}_2= \{F \in CB(X): rad_Y(F)\leq \alpha\}.\)
\((Y, \mathcal{F}_3)\) has property-\((P_2)\), where \(\mathcal{F}_3= \{F \in CB(X): rad_Y(F)=\alpha\}.\)
Proof. \((1)\Rightarrow (2)\): Let \(\epsilon >0.\) By \((1),\) there exists \(0<\delta <1\) such that
\(Z_Y(G,\delta) \subseteq Z_Y(G)+\frac{\epsilon}{\alpha}B_X\) for every \(G \in \mathcal{F}_1.\) Now for \(\delta_1 = \frac{\epsilon\delta}{3},\) we show
that \(Z_Y(F, \delta_1) \subseteq Z_Y(F)+\epsilon B_X\) for every \(F\in\mathcal{F}_2.\) Let \(F \in \mathcal{F}_2\).
Case-(i): Let \(rad_Y(F) \leq \frac{\epsilon}{3}.\) Then for any \(v \in Z_Y(F, \delta_1)\) and \(w \in Z_Y(F)\) we have \[\|v-w\|
\leq r(v, F)+r(w, F) \leq rad_Y(F)+\delta_1+rad_Y(F) < \epsilon.\] Thus, \(Z_Y(F,\delta_1) \subseteq Z_Y(F)+\epsilon B_X.\)
Case-(ii): Let \(\frac{\epsilon}{3}< rad_Y(F) \leq \alpha.\) Consider \(G=\frac{1}{rad_Y(F)}(F-w),\) for some \(w \in Z_Y(F).\) Observe that \(rad_Y(G)=r(0, G)=1.\) Thus, by assumption, we have \(Z_Y(G,\delta) \subseteq Z_Y(G)+\frac{\epsilon}{\alpha}B_X.\) For any \(\eta\geq 0\), it is easy to verify
that \[Z_Y(G, \eta) = \frac{1}{rad_Y(F)} \left(Z_Y(F, rad_Y(F)\eta)-w \right).\] Therefore, we have \(Z_Y(F, rad_Y(F)\delta) \subseteq Z_Y(F)+rad_Y(F)\frac{\epsilon}{\alpha}B_X\) and hence
it follows that \(Z_Y(F,\delta_1) \subseteq Z_Y(F)+\epsilon B_X.\)
\((2)\Rightarrow(3)\): Obvious.
\((3)\Rightarrow(1)\): Let \(\epsilon>0.\) By (3), there exists \(\delta>0\) such that \(Z_Y(G,\alpha\delta)\subseteq
Z_Y(G)+\alpha\epsilon B_X\) for every \(G \in \mathcal{F}_3.\) Let \(F\in \mathcal{F}_1.\) Since \(\alpha F \in \mathcal{F}_3,\) we have \(Z_Y(\alpha F, \alpha \delta) \subseteq Z_Y(\alpha F)+\alpha \epsilon B_X\), which further leads to \(Z_Y(F, \delta) \subseteq Z_Y(F)+\epsilon B_X.\) Hence, \((Y,
\mathcal{F}_1)\) has property-\((P_2).\) ◻
Now, we present some results which exhibit equivalent relations between property-\((P_2)\) and geometric properties.
As a sufficient condition for the existence of restricted centers, Calder et al. [9] introduced the following notion. The closed ball centered at \(x \in X\) with radius \(r>0\) is denoted by \(B[x,r]\).
Definition 2. [9] Let \(Y\) be a subspace of \(X\). We say \(X\) is quasi uniformly rotund with respect to \(Y\), if for every \(\epsilon>0\) there exists \(\delta>0\) such that for every \(v \in Y\), there exists \(w \in Y\) such that \(w \in B[0, \epsilon]\) and \(B[0,1] \cap B[v,1-\delta] \subseteq B[w, 1-\delta]\).
For more results on quasi uniform rotundity in terms of restricted centers, we refer to [2], [8], [10]. In [2], it is shown that property-\((P_2)\) is a necessary condition for quasi uniform rotundity. However, in 4, we show that the converse also holds through the continuity of the restricted center map. For this, we recall the following continuity definitions.
Definition 3. Let \((N,d)\) be a metric space, \(X\) be a Banach space and \(T: (N,d) \to (CB(X), \mathcal{H})\) be a set valued map. Then the map \(T\) is said to be
upper Hausdorff semi-continuous (in short, uHsc) on \(N\) if for every \(a_0 \in N\) and \(\epsilon>0\) there exists \(\delta>0\) such that \(T(a) \subseteq T(a_0)+ \epsilon B_X\), whenever \(a \in N\) and \(d(a, a_0) < \delta\);
lower Hausdorff semi-continuous (in short, lHsc) on \(N\) if for every \(a_0 \in N\) and \(\epsilon>0\) there exists \(\delta>0\) such that \(T(a_0) \subseteq T(a)+ \epsilon B_X\), whenever \(a \in N\) and \(d(a, a_0) < \delta\);
Hausdorff continuous on \(N\) if it is both uHsc and lHsc on \(N\);
uniformly Hausdorff continuous on \(N\) if for every \(\epsilon>0\) there exists \(\delta>0\) such that \(\mathcal{H}(T(a), T(b))< \epsilon\), whenever \(a,b \in N\) with \(d(a, b) < \delta\).
In the above definition, if \(T(a)\) is a singleton set for every \(a \in N,\) then upper Hausdorff semi-continuity, lower Hausdorff semi-continuity and Hausdorff continuity of \(T\) are equivalent on \(N\).
Theorem 4. Let \(Y\) be a subspace of \(X\) and \(\mathcal{F}=\{F \in CB(X): rad_{Y}(F) \leq \alpha\}\) for some \(\alpha>0.\) Then the following statements are equivalent.
\(X\) is quasi uniformly rotund with respect to \(Y\).
\((Y, \mathcal{F})\) has property-\((P_2)\).
The restricted center map \(Z_Y\) is non-empty valued and uniformly Hausdorff continuous on \((\mathcal{F}, \mathcal{H}).\)
Proof. \((1) \Rightarrow (2)\): This implication follows from [2] and 3.
\((2) \Rightarrow (3)\): This implication follows from [2].
\((3) \Rightarrow (1)\): This implication follows from [10]. ◻
Now, we present a characterization of uniform rotundity with respect to a subspace in terms of property-\((P_2)\), which is not explicitly stated in the literature; however, it follows from the results of [2], [11], and this characterization will be utilized in 3.
Definition 4. [11] Let \(Y\) be a subspace of \(X\). We say \(X\) is uniformly rotund with respect to \(Y\) if \((x_n)\) and \((y_n)\) are two sequences in \(S_X\) such that \(x_n-y_n \in Y\) for every \(n \in \mathbb{N}\) and \(\left\Vert \frac{x_n+y_n}{2} \right\Vert \to 1,\) it follows that \(x_n-y_n \to 0\).
Theorem 5. Let \(Y\) be a subspace of \(X\) and \(\alpha>0\). Then the following statements are equivalent.
\(X\) is uniformly rotund with respect to \(Y.\)
\((Y, \mathcal{F}_1)\) has property-\((P_2)\) and \(Z_Y(F)\) is singleton for every \(F \in \mathcal{F}_1\), where \(\mathcal{F}_1=\{F \in CB(X): rad_Y(F) \leq \alpha\}\).
\((Y, \mathcal{F}_2)\) has property-\((P_2)\) and \(Z_Y(F)\) is singleton for every \(F \in \mathcal{F}_2\), where \(\mathcal{F}_2=\{F \in K(X): rad_Y(F) \leq \alpha\}\).
Proof. \((1) \Leftrightarrow (2)\): These implications follow from [2].
\((2) \Rightarrow (3)\): Obvious.
\((3) \Rightarrow (1)\): Since, by [2], the restricted center map \(Z_Y\) is uniformly
Hausdorff continuous on \((\mathcal{F}_2, \mathcal{H}),\) it follows from the assumption and [11], that \(X\) is uniformly rotund with respect to \(Y\). ◻
This section deals with the stability of property-\((P_1),\) property-\((P_2)\) and continuity of the restricted center map under \(\ell_p\)-direct sum \((1 \leq p < \infty).\) We need the following sequential characterizations of properties-\((P_1)\) and \((P_2)\) which are easy to verify.
Proposition 6. Let \(V \in CL(X)\), \(F \in CB(X)\) and \(\mathcal{F} \subseteq CB(X)\). Then the following statements hold.
\((V, F)\) has property-\((P_1)\) if and only if \((V, F)\) has r.c.p. and for every minimizing sequence \((x_n)\) for \(F\) in \(V\), there exists a sequence \((y_n)\) in \(V\) such that \(y_n \in Z_V(F)\) for every \(n \in \mathbb{N}\) satisfying \(\|x_n-y_n\| \to 0.\)
\((V, \mathcal{F})\) has property-\((P_2)\) if and only if \((V, \mathcal{F})\) has r.c.p. and whenever \((x_n)\) is a sequence in \(V\), \((F_n)\) is a sequence in \(\mathcal{F}\) with \(r(x_n,F_n)- rad_V(F_n) \to 0\), there exists a sequence \((y_n)\) in \(V\) such that \(y_n \in Z_V(F_n)\) for every \(n \in \mathbb{N}\) satisfying \(\|x_n-y_n\| \to 0.\)
To see the stability of property-\((P_1)\), we need the following lemma, which is a generalization of [7].
Lemma 1. Let \(\{X_i : {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p <\infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}},\) \(Y=(\oplus_pY_i)_{i \in \mathbb{N}},\) \(F=\Pi_{i \in \mathbb{N}}F_i \in \mathcal{P}(X)\) and \((y_n)\) be a minimizing sequence for \(F\) in \(Y\), where \(y_n=(y_{n,i})_{i \in \mathbb{N}}\) for every \(n \in \mathbb{N}.\) Then for every \(\epsilon >0\) there exists \(j \in \mathbb{N}\) such that \(\sum_{i>j}\|y_{n,i}\|^p < \epsilon^p\) for all \(n \in \mathbb{N}.\)
Proof. Suppose there exists \(\epsilon >0\) such that for every \(j \in \mathbb{N}\) there exists a subsequence \((n_m)\), depending on \(j\), such that \(\sum_{i>j}\|y_{n_m, i}\|^p \geq \epsilon^p\) for all \(m \in \mathbb{N}.\) Since, by 1, \(r(0, F)^p = \sum_{i \in \mathbb{N}} r(0, F_i)^p,\) there exists \(j_0 \in \mathbb{N}\) such that \(\sum_{i>j_0} r(0, F_i)^p < (\frac{\epsilon}{3})^p.\) By preceding argument, for this ‘\(j_0\)’ there exists a subsequence \((n_k)\) such that \(\sum_{i>j_0} \|y_{n_k, i}\|^p \geq \epsilon^p\) for all \(k \in \mathbb{N}.\) For any \(k \in \mathbb{N},\) by 1, we have \[\begin{align} r(y_{n_k}, F)^p & = \left(\sum_{i \leq j_0} r(y_{n_k,i}, F_i)^p +\sum_{i>j_0} r(0, F_i)^p \right) -\sum_{i>j_0} r(0, F_i)^p+ \sum_{i >j_0} r(y_{n_k,i}, F_i)^p \\ & > rad_Y(F)^p - \left(\frac{\epsilon}{3}\right)^p + \left[\left(\sum_{i>j_0} \|y_{n_k,i}\|^p\right)^{\frac{1}{p}}-\left(\sum_{i>j_0} r(0, F_i)^p\right)^{\frac{1}{p}}\right]^p \\ & > rad_Y(F)^p - \left(\frac{\epsilon}{3}\right)^p +\left(\epsilon-\frac{\epsilon}{3}\right)^p\\ & \geq rad_Y(F)^p + \left(\frac{\epsilon}{3}\right)^p. \end{align}\] This is a contradiction. Hence the proof. ◻
Theorem 7. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}},\) \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\) and \(F=\Pi_{i \in \mathbb{N}}F_i \in \mathcal{P}(X).\) Then \((Y_i, F_i)\) has property-\((P_1)\) for every \(i\in \mathbb{N}\) if and only if \((Y, F)\) has property-\((P_1).\)
Proof. Let \((Y_i, F_i)\) has property-\((P_1)\) for every \(i\in \mathbb{N}\). By 2, \((Y, F)\) has r.c.p. Let \((y_n)\) be a minimizing sequence for \(F\) in \(Y\), where \(y_n=(y_{n,i})_{i \in \mathbb{N}}\) for all \(n \in \mathbb{N}.\) Let \(i \in \mathbb{N}\). By 1, we have \[\begin{align} rad_{Y_i}(F_i)^p &\leq r(y_{n, i}, F_i)^p \\ & = r(y_n, F)^p -\sum_{j \in \mathbb{N} \backslash \{i\}}r(y_{n,j}, F_j)^p \\ & \leq r(y_n, F)^p -\sum_{j \in \mathbb{N} \backslash \{i\}} rad_{Y_j}(F_j)^p + rad_Y(F)^p-rad_Y(F)^p \\ & \leq r(y_n, F)^p-rad_Y(F)^p+ rad_{Y_i}(F_i)^p. \end{align}\] Thus, \(r(y_{n,i}, F_i) \to rad_{Y_i}(F_i).\) Since \((Y_i, F_i)\) has property-\((P_1),\) by 6, there exists \(z_{n,i} \in Z_{Y_i}(F_i)\) for every \(n \in \mathbb{N}\) such that \(\|y_{n, i}-z_{n, i}\| \to 0\) as \(n \to \infty.\) By 2, we have \(z_n=(z_{n,i})_{i\in \mathbb{N}} \in Z_Y(F)\) for every \(n \in \mathbb{N}.\) Now, by 6, it is enough to prove that \(\|y_n-z_n\|\to 0.\) To see this let \(\epsilon >0.\) By 1, there exists \(j \in \mathbb{N}\) such that \(\sum_{i>j}\|y_{n,i}\|^p < \epsilon^p\) and \(\sum_{i>j}\|z_{n,i}\|^p < \epsilon^p\) for all \(n \in \mathbb{N}\). Also, there exists \(n_0 \in \mathbb{N}\) such that \(\sum_{i \leq j}\|y_{n, i}-z_{n,i}\|^p \leq (2\epsilon)^p\) for all \(n \geq n_0.\) Therefore for any \(n \geq n_0\), we have \[\begin{align} \|y_n-z_n\|^p & = \sum_{i \leq j}\|y_{n,i}-z_{n,i}\|^p + \sum_{i>j} \|y_{n,i}-z_{n,i}\|^p \\ & \leq (2\epsilon)^p + \left(\left(\sum_{i > j}\|y_{n,i}\|^p\right)^\frac{1}{p}+ \left(\sum_{i > j}\|z_{n,i}\|^p\right)^\frac{1}{p}\right)^p\\ & < (2\epsilon)^p + (2\epsilon)^p. \end{align}\] Thus, \(\|y_n-z_n\| \to 0.\) To prove the converse, let \((Y, F)\) has property-\((P_1)\). It follows from 2 that \((Y_i, F_i)\) has r.c.p for every \(i \in \mathbb{N}\). Let \(j \in \mathbb{N}\) and \((y_{n,j})_{n \in \mathbb{N}}\) be a minimizing sequence for \(F_{j}\) in \(Y_{j}\). For every \(i \in \mathbb{N}\), choose \(y_i \in Z_{Y_i}(F_i)\). Now for every \(n \in \mathbb{N},\) consider \(y_{n,i}= y_i\) for every \(i \in \mathbb{N} {\setminus} \{j\}\) and \(z_n=(y_{n,i})_{i \in \mathbb{N}} \in Y.\) By 1, it follows that \[rad_Y(F)^p \leq r(z_n,F)^p = \sum_{i \in \mathbb{N}} r(y_{n,i}, F_i)^p = \sum_{i \in \mathbb{N} {\setminus} \{j\}} rad_{Y_i}(F_i)^p+ r(y_{n, j}, F_{j})^p\] for all \(n \in \mathbb{N}\) and hence \(r(z_n,F) \to rad_Y(F).\) Since \((Y, F)\) has property-\((P_1),\) by 6, there exists \(w_n=(w_{n,i})_{i \in \mathbb{N}} \in Z_Y(F)\) for every \(n \in \mathbb{N}\) such that \(\|z_n-w_n\| \to 0\), which further implies \(\|y_{n,j}-w_{n,j}\| \to 0\). Note that, by 2, \(w_{n,j} \in Z_{Y_{j}}(F_{j})\) for all \(n \in \mathbb{N}.\) Thus, \((Y_{j}, F_{j})\) has property-\((P_1)\). Hence the proof. ◻
The following corollary is a direct consequence of 7.
Corollary 2. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}.\) Then \(Y_i\) is strongly proximinal on \(X_i\) for every \(i\in \mathbb{N}\) if and only if \(Y\) is strongly proximinal on \(X\).
For any \(V \in CL(X)\) and \(\mathcal{F} \subseteq CB(X),\) we say that the pair \((V, \mathcal{F})\) is well posed (respectively, generalized well posed) if every minimizing sequence for \(F \in \mathcal{F}\) in \(V\) converges (respectively, has a convergent subsequence). The subsequent result on the stability of well posed-ness and generalized well posed-ness follows using 2 and a similar technique involved in the proof of 7.
Corollary 3. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\). Then \((Y_i, CB(X_i))\) is well posed (respectively, generalized well posed) for every \(i\in \mathbb{N}\) if and only if \((Y, \mathcal{P}(X))\) is well posed (respectively, generalized well posed).
Now, we proceed to discuss the stability of property-\((P_2)\). It is easy to verify from the proof of 7 that if \((Y, \mathcal{F})\) has property-\((P_2)\), where \(\mathcal{F}=\{F \in \mathcal{P}(X): rad_Y(F)=1\}\), then \((Y_i, \mathcal{F}_i)\) has property-\((P_2)\), where \(\mathcal{F}_i=\{F_i \in CB(X_i): rad_{Y_i}(F_i)=1\}\) for every \(i\in \mathbb{N}\). However, the following example reveals that, for any \(1 \leq p < \infty\), the converse does not hold for infinite collection. Nevertheless, in the subsequent result, we prove that the converse is true when the collection is finite.
Example 1. Let \(1\leq p < \infty\). For every \(i \in \mathbb{N}\), let \(X_i= (\mathbb{R}^3, \|\cdot\|_{p_i})\), where \(p_i=1+\frac{1}{i}\) and \(Y_i=span\{(1,0,0), (0,1,0)\}\) be the subspace of \(X_i.\) Consider \(X=(\oplus_p X_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_p Y_i)_{i \in \mathbb{N}}.\) Since each \(X_i\) is uniformly rotund, by [2], it follows that \((Y_i, \mathcal{F}_i)\) has property-\((P_2)\) for every \(i \in \mathbb{N}\), where \(\mathcal{F}_i=\{F_i \in CB(X_i): rad_{Y_i}(F_i)=1\}.\) Now we prove that \((Y, \mathcal{F})\) does not have property-\((P_2)\), where \(\mathcal{F}=\{F\in \mathcal{P}(X): rad_Y(F)=1\}.\) Let \(n \in \mathbb{N}.\) Consider \(x_n= (x_{n,i})_{i \in \mathbb{N}} \in Y\) and \(y_n= (y_{n,i})_{i \in \mathbb{N}} \in Y\) where \[\begin{align} x_{n,i}= \begin{cases}(0,0,0), & \text{if } i\neq n ;\\ (1,0,0), & \text{if } i=n, \end{cases}&y_{n,i}= \begin{cases}(0,0,0), & \text{if } i\neq n ;\\ (0,1,0), & \text{if } i=n \end{cases} \end{align}\] and \(G_n=\left\{\frac{x_n+y_n}{\|x_n+y_n\|}, -\frac{x_n+y_n}{\|x_n+y_n\|}\right\} \in CB(X).\) Since \(G_n= \Pi_{i \in \mathbb{N}} G_{n,i}\) where \[G_{n,i}=\begin{cases}\{(0,0,0)\}, & \text{if } i\neq n ;\\ \left\{ \frac{(1,1,0)}{\|(1,1,0)\|_{p_n}}, \frac{-(1,1,0)}{\|(1,1,0)\|_{p_n}}\right\}, & \text{if } i=n, \end{cases}\] we have \(G_n \in \mathcal{P}(X).\) Observe that \(rad_Y(G_{n})=r(0, G_n)= 1.\) Define \(w_n=\frac{x_n-y_n}{\|x_n+y_n\|} \in Y.\) Since \[rad_Y(G_n) \leq r(w_n,G_n) = \max\left\{\left\Vert w_n+\frac{x_n+y_n}{\|x_n+y_n\|}\right\Vert, \left\Vert w_n-\frac{x_n+y_n}{\|x_n+y_n\|}\right\Vert \right\}=\frac{2}{\|x_n+y_n\|}\] for all \(n \in \mathbb{N}\) and \(\left\Vert\frac{x_n+y_n}{2}\right\Vert \to 1\), we have \(r(w_n, G_n) -rad_Y(G_n) \to 0\). Now, by 6, it is enough to prove that \(d(w_n, Z_Y(G_n))\) does not converge to \(0\). Fix \(n \in \mathbb{N}\). Since for any \(i \in \mathbb{N}\), \(X_i\) is uniformly rotund, by [12], it follows that \(Z_{Y_i}(G_{n,i})\) is singleton. Observe that \(Z_{Y_i}(G_{n,i})=\{0\}\) for all \(i \in \mathbb{N}\) further, using 2, we have \(Z_Y(G_n)=\{0\}.\) Thus, \[d(w_n, Z_Y(G_n))=\|w_n-0\|= \left\Vert \frac{x_n-y_n}{\|x_n+y_n\|} \right\Vert=1.\] Hence, \((Y, \mathcal{F})\) does not have property-\((P_2)\).
Theorem 8. Let \(\{X_i :{i\in I}\}\) be a finite collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in I\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in I}\) and \(Y=(\oplus_pY_i)_{i \in I}.\) Then \((Y_i, \mathcal{F}_i)\) has property-\((P_2)\) for every \(i\in I\), where \(\mathcal{F}_i=\{F_i \in CB(X_i): rad_{Y_i}(F_i)= 1\}\) if and only if \((Y, \mathcal{F})\) has property-\((P_2)\), where \(\mathcal{F}=\{F \in \mathcal{P}(X): rad_Y(F) = 1\}\).
Proof. It is enough to prove the necessary part. Clearly, by 2, \((Y, \mathcal{F})\) has r.c.p. Let \((G_n)\) be a sequence in \(\mathcal{F}\) and \((w_n)\) be a sequence in \(Y\) such that \(r(w_n, G_n) - rad_Y(G_n) \to 0,\) where \(G_n= \Pi_{i \in I} G_{n,i}\) and \(w_n=(w_{n,i})_{i \in I}\) for every \(n \in \mathbb{N}\). Let \(i \in I.\) Since \[\begin{align} 0 &\leq r(w_{n, i}, G_{n,i})^p- rad_{Y_i}(G_{n,i})^p\\ & = r(w_{n, i}, G_{n,i})^p + \sum_{j\in I {\setminus} \{i\}} rad_{Y_j}(G_{n,j})^p- \left(rad_{Y_i}(G_{n,i})^p + \sum_{j\in I {\setminus} \{i\}} rad_{Y_j}(G_{n,j})^p\right)\\ & \leq r(w_n, G_n)^p-rad_Y(G_n)^p, \end{align}\] we have \(r(w_{n,i}, G_{n,i})- rad_{Y_i}(G_{n,i}) \to 0\) as \(n \to \infty\). Since \(rad_{Y_i}(G_{n,i}) \leq 1\), by assumption and 3, there exists \(z_{n,i} \in Z_{Y_i}(G_{n,i})\) for every \(n \in \mathbb{N}\) such that \(\|w_{n, i}-z_{n, i}\| \to 0\) as \(n \to \infty.\) Now for every \(n \in \mathbb{N},\) consider \(z_n=(z_{n,i})_{i\in I} \in Y\) and observe that, by 2, \(z_n \in Z_Y(G_n)\). Clearly, we have \(\|w_n-z_n\| \to 0.\) Thus, \((Y, \mathcal{F})\) has property-\((P_2)\). ◻
The proof of the next result follows in the similar lines of the proof of 8.
Corollary 4. Let \(\{X_i :{i\in I}\}\) be a finite collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in I\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in I}\) and \(Y=(\oplus_pY_i)_{i \in I}.\) Then \(Y_i\) is uniformly strongly proximinal on \(Y_i^\prime\) for every \(i\in I\), where \(Y'_i=\{x_i \in X_i: d(x_i, Y_i) =1\}\) if and only if \(Y\) is uniformly strongly proximinal on \(Y^\prime\), where \(Y'=\{x \in X: d(x, Y) =1\}\).
Let \(V \in CL(X)\) and \(\mathcal{F} \subseteq CB(X)\). Recall that if \((V, \mathcal{F})\) has property-\((P_1),\) then the restricted center map \(Z_V\) is uHsc on \((\mathcal{F}, \mathcal{H})\). Also if \((V, \mathcal{F})\) has property-\((P_2),\) then the map \(Z_V\) is uniformly Hausdorff continuous on \((\mathcal{F}, \mathcal{H})\). It is interesting to check the existence of an intermediate notion that provides the Hausdorff continuity of restricted center map. For this, we introduce the following notion.
Definition 5. Let \(V \in CL(X)\) and \(\mathcal{F} \subseteq CB(X).\) We say that the pair \((V, \mathcal{F})\) has property-\((lP_2)\), if \((V, \mathcal{F})\) has r.c.p. and for every \(\epsilon>0\), \(F \in \mathcal{F}\) there exists \(\delta>0\) such that \(Z_V(G, \delta) \subseteq Z_V(G)+\epsilon B_X\) whenever \(G \in \mathcal{F}\) and \(\mathcal{H}(F, G)< \delta.\)
Note that, in the above definition, if \(\mathcal{F}\) is a collection of all singleton subsets of \(X\), then property-\((lP_2)\) is equivalent to locally uniform strong proximinality [4], i.e., \((V, \mathcal{F})\) has property-\((lP_2)\) if and only if \(V\) is locally uniformly strongly proximinal on \(X\). Clearly, \((V, \mathcal{F})\) has property-\((P_2) \Rightarrow (V, \mathcal{F})\) has property-\((lP_2) \Rightarrow (V, \mathcal{F})\) has property-\((P_1)\). However, the reverse implications are not true in general (see, 2).
In the following result, we establish a relation between property-\((P_1)\) and property-\((lP_2).\) This relation exhibits that property-\((lP_2)\) is sufficient for the Hausdorff continuity of the restricted center map.
Theorem 9. Let \(V \in CL(X)\) and \(\mathcal{F} \subseteq CB(X).\) Then the following statements are equivalent.
\((V, \mathcal{F})\) has property-\((lP_2).\)
For every \(\epsilon>0\) and \(F \in \mathcal{F}\) there exists \(\delta>0\) such that \(Z_V(F, \delta) \subseteq Z_V(G)+ \epsilon B_X\) whenever \(G \in \mathcal{F}\) and \(\mathcal{H}(F,G)< \delta.\)
\((V, \mathcal{F})\) has property-\((P_1)\) and the map \(Z_V\) is lHsc on \((\mathcal{F}, \mathcal{H})\).
Proof. \((1) \Rightarrow (2)\): Let \(\epsilon>0\) and \(F \in \mathcal{F}\). By \((1)\), there exists \(\delta>0\) such that \(Z_V(G, \delta) \subseteq Z_V(G)+ \epsilon B_X\) whenever \(G \in \mathcal{F}\) and \(\mathcal{H}(F,G)<
\delta.\) Choose \(\delta_1= \frac{\delta}{3}\). Let \(G \in \mathcal{F}\) and \(\mathcal{H}(F,G)< \delta_1\). Since for any \(v \in Z_V(F, \delta_1)\) we have \[r(v, G) \leq r(v, F)+\mathcal{H}(F, G) < rad_V(F)+2\delta_1 < rad_V(G)+3\delta_1,\] it follows that \(Z_V(F, \delta_1) \subset
Z_V(G, \delta).\) Therefore, we have \(Z_V(F, \delta_1) \subseteq Z_V(G)+ \epsilon B_X\).
\((2) \Rightarrow (3)\): Obvious.
\((3) \Rightarrow (1)\): Let \(\epsilon >0\) and \(F \in \mathcal{F}.\) By our assumption, there exists \(\delta'
>0\) such that \(Z_V(F, \delta') \subseteq Z_V(F)+\frac{\epsilon}{2} B_X\) and \(Z_V(F) \subseteq Z_V(A)+\frac{\epsilon}{2} B_X\) for every \(A \in
\mathcal{F}\) with \(\mathcal{H}(F, A)<\delta'.\) For \(\delta =\frac{\delta^\prime}{3},\) we claim that \(Z_V(G, \delta) \subseteq Z_V(G)+\epsilon
B_X\) for any \(G\in \mathcal{F}\) with \(\mathcal{H}(F, G)<\delta.\) Let \(v \in Z_V(G, \delta).\) Since \[r(v, F) \leq
r(v, G)+\mathcal{H}(F, G) < rad_V(G)+2\delta < rad_V(F)+3\delta,\] we have \(v \in Z_V(F,\delta^\prime).\) Thus, \[Z_V(G, \delta) \subseteq Z_V(F, \delta^\prime) \subseteq
Z_V(F)+\frac{\epsilon}{2} B_X \subseteq Z_V(G)+\epsilon B_X.\] Hence, \((V, \mathcal{F})\) has property-\((lP_2).\) ◻
Corollary 5. Let \(V \in CL(X)\). Then \(V\) is strongly proximinal on \(X\) and the map \(P_V\) is lHsc on \(X\) if and only if \(V\) is locally uniformly strongly proximinal on \(X\).
The following example reveals that the property-\((lP_2)\) is stronger than property-\((P_1)\) but weaker to property-\((P_2).\) We say a space \(X\) is uniformly rotund in every direction (in short, URED) [11], if \(X\) is uniformly rotund with respect to every one dimensional subspace of \(X\).
Example 2.
Consider \(X=(\mathbb{R}^3, \|\cdot\|_c),\) where \(\|x\|_c=|x_1|+(x_2^2+ x_3^2)^{\frac{1}{2}}\) for any \(x=(x_1, x_2, x_3) \in \mathbb{R}^3\) and \(Y= span \{(1,1,0)\}.\) Note that, from [13], the metric projection \(P_Y\) is not lHsc on \(X.\) Thus, \(Z_Y\) is not lHsc on \(CB(X).\) Therefore, by 9, \((Y,CB(X))\) does not have property-\((lP_2).\) However, by [1], \((Y, CB(X))\) has property-\((P_1).\)
Consider a rotund space \(X\) which is not URED (see, [14]). Then there exists a one dimensional subspace \(Y\) of \(X\) such that \(X\) is not uniformly rotund with respect to \(Y.\) By [1], \((Y,\mathcal{F})\) has property-\((P_1)\), where \(\mathcal{F}=\{F \in K(X): rad_Y(F)=1\}.\) Since \(X\) is rotund, by [12], \(Z_Y(A)\) is singleton for every \(A \in \mathcal{F}.\) Thus, \(Z_Y\) is Hausdorff continuous on \((\mathcal{F}, \mathcal{H}).\) Therefore, by 9, \((Y,\mathcal{F})\) has property-\((lP_2).\) However, from 5, it follows that \((Y,\mathcal{F})\) does not have property-\((P_2).\)
Whenever \(X\) is rotund, \(V \in CC(X)\) and \(\mathcal{F} \subseteq K(X),\) it follows from [11] and 9 that \((V, \mathcal{F})\) has property-\((P_1)\) if and only if \((V, \mathcal{F})\) has property-\((lP_2)\). However, in 2, notice that even in finite dimensional spaces, property-\((P_1)\) and property-\((lP_2)\) are not equivalent on \(CB(X).\) In the following result, we observe that property-\((P_1)\) and property-\((lP_2)\) coincide for a large class of spaces.
Proposition 10. Let \(X\) be a URED space. Then the following statements hold.
Let \(C \in CC(X)\), \(\mathcal{F} \subseteq CB(X)\). Then \((C,\mathcal{F})\) has property-\((P_1)\) if and only if \((C,\mathcal{F})\) has property-\((lP_2).\)
Let \(Y\) be any finite dimensional subspace of \(X\). Then \((Y, \mathcal{F}')\) has property-\((P_2)\), where \(\mathcal{F}'=\{F \in CB(X): rad_Y(F)=1\}.\)
Proof. \((1)\): Let \((C,\mathcal{F})\) has property-\((P_1)\). By [12], \(Z_C(A)\) is singleton for every \(A \in \mathcal{F}.\) Therefore, by [1] and 9, \((C,\mathcal{F})\) has property-\((lP_2).\)
\((2)\): Since \(X\) is URED and \(dim(Y)< \infty\), by [15], it follows that \(X\) is uniformly rotund with respect to \(Y\). Therefore, by 5, \((Y, \mathcal{F}')\) has property-\((P_2)\). ◻
In general, for hyperplane \(H\) of \(X\), from [3] it follows that the notions strong proximinality, locally uniform strong proximinality and uniform strong proximinality coincide. However the following example reveals that the properties-\((P_1)\), \((lP_2)\) and \((P_2)\) do not coincide for hyperplanes. We say that a Banach space \(X\) has Kadets-Klee property if \(x \in S_X\) and \((x_n)\) is a sequence in \(S_X\) with \(x_n \xrightarrow{w} x\), it follows that \(x_n \to x.\)
Example 3.
Let \(W\) be any finite dimensional space such that the center map \(Z_W\) is not Hausdorff continuous on \(CB(W)\) [16]. Consider \(X=W \oplus_2 \mathbb{R}\) and \(Y= W \oplus_2 \{0\}.\) Clearly, the restricted center map \(Z_Y\) is not Hausdorff continuous on \(CB(X)\). Therefore, by 9, \((Y, CB(X))\) does not have property-\((lP_2).\) However, by [1], \((Y, CB(X))\) has property-\((P_1).\)
Consider a non uniformly rotund space \(M\) which is rotund, reflexive and has Kadets-Klee property (see, [14]). Let \(X= M \oplus_2 \mathbb{R}\) and \(H=M \oplus_2 \{0\}.\) Clearly, \(X\) is rotund, reflexive and has Kadets-Klee property. Thus, by [2] and [11], it follows that \((H, K(X))\) has property-\((P_1)\) and \(Z_H(A)\) is singleton for every \(A \in K(X).\) Further, by [1] and 9, \((H, K(X))\) has property-\((lP_2).\) Since \(X\) is not uniformly rotund with respect to \(H,\) by 5, \((H, \mathcal{F})\) does not have property-\((P_2)\), where \(\mathcal{F}=\{F \in K(X): rad_H(F) \leq 1\}\).
[eg:_p1_lp1_p2,eg: lP_2 P_2] further reveals that, in general, properties-\((lP_2)\) and \((P_2)\) do not coincide in infinite dimensional spaces even when the subspaces are finite dimensional or finite co-dimensional. However, the following result illustrates that in finite dimensional spaces, the properties-\((lP_2)\) and \((P_2)\) coincide.
Proposition 11. Let \(V \in CL(X)\) and \(\mathcal{F}\) be a compact subset of \((CB(X), \mathcal{H})\). Then \((V, \mathcal{F})\) has property-\((lP_2)\) if and only if \((V, \mathcal{F})\) has property-\((P_2)\).
Proof. It is enough to prove the necessary part. Let \(\epsilon>0\). By assumption, for every \(F \in \mathcal{F}\) there exists \(\delta_F>0\) such that \(Z_V(G, \delta_F) \subseteq Z_V(G)+ \epsilon B_X\) whenever \(G \in \mathcal{F}\) and \(\mathcal{H}(F,G)< \delta_F.\) Since \(\{ B(F, \delta_F): F \in \mathcal{F}\}\) is an open cover for \(\mathcal{F}\), there exist \(F_1, F_2, \dots, F_n \in \mathcal{F}\) such that \(\mathcal{F} \subseteq \cup_{i=1}^{n} B(F_i, \delta_{F_i})\). Choose \(\delta= \min\{\delta_{F_i}: 1 \leq i \leq n\}.\) Let \(F \in \mathcal{F}.\) Then \(F \in B(F_j, \delta_{F_j})\) for some \(1 \leq j \leq n.\) Therefore, \(Z_V(F, \delta) \subseteq Z_V(F, \delta_{F_j}) \subseteq Z_V(F)+ \epsilon B_X.\) Hence the proof. ◻
Corollary 6. Let \(X\) be a finite dimensional space and \(Y\) be a subspace of \(X\). Then \((Y, CB(X))\) has property-\((lP_2)\) if and only if \((Y, \mathcal{F})\) has property-\((P_2)\), where \(\mathcal{F}=\{F \in CB(X): rad_Y(F)=1\}\).
Proof. Let \((Y, CB(X))\) has property-\((lP_2)\). By 3, it is enough to prove that \((Y, \mathcal{F}')\) has property-\((P_2)\), where \(\mathcal{F}'=\{F \in CB(X): rad_Y(F)= r(0,F)=1\}\). Observe that \(\mathcal{F}'\) is a closed subset of \(CB(B_X).\) Since \(B_X\) is compact, by [17], \((CB(B_X), \mathcal{H})\) is compact subset of \((CB(X), \mathcal{H})\). Therefore, \(\mathcal{F}'\) is compact subset of \((CB(X), \mathcal{H})\). Thus, by 11, \((Y, \mathcal{F}')\) has property-\((P_2)\). The converse is easy to verify using 3. Hence the proof. ◻
The following corollary is a consequence of [lem: QUR,thrm: P_1 lP_2 lHsc], and 6. The result also reveals that the lower Hausdorff semi-continuity of restricted center map is sufficient for the quasi uniform rotundity whenever the space is of finite dimension.
Corollary 7. Let \(X\) be a finite dimensional space and \(Y\) be a subspace of \(X\). If \(Z_Y\) is lHsc on \((CB(X), \mathcal{H})\), then \(Z_Y\) is uniformly Hausdorff continuous on \((\mathcal{F}, \mathcal{H})\), where \(\mathcal{F}=\{F \in CB(X): rad_Y(F)\leq 1\}\). In particular, \(X\) is quasi uniformly rotund with respect to \(Y\).
In view of [thm:pds_P1,thm:pfds_P2] and 1, it is reasonable to investigate the stability of property-\((lP_2)\) under \(\ell_p\)-direct sum. To examine this, it is enough to check, by 9, the stability of continuity properties of the restricted center map.
Theorem 12. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) such that \((Y_i,CB(X_i))\) has r.c.p. for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\). Then \(Z_{Y_i}\) is lHsc (respectively, uHsc) on \(CB(X_i)\) for every \(i\in \mathbb{N}\) if and only if \(Z_Y\) is lHsc (respectively, uHsc) on \(\mathcal{P}(X)\).
Proof. We present the proof for lHsc, the proof of uHsc follows in similar lines. Let \(Z_{Y_i}\) be lHsc on \(CB(X_i)\) for every \(i \in \mathbb{N}.\) Let \(F= \Pi_{i \in \mathbb{N}} F_i \in \mathcal{P}(X)\), \(G_n = \Pi_{i \in \mathbb{N}} G_{n,i} \in \mathcal{P}(X)\) for every \(n \in \mathbb{N}\) satisfying \(\mathcal{H}(G_n,F) \to 0\) and \((y_n)\) be a sequence in \(Z_Y(F)\) where \(y_n= (y_{n,i})_{i \in \mathbb{N}}\). We need to prove that \(d(y_n, Z_Y(G_n)) \to 0.\) Let \(i \in \mathbb{N}\). Observe that \(\mathcal{H}(G_{n,i},F_i) \leq \mathcal{H}(G_n, F)\) holds for every \(n \in \mathbb{N}\), hence \(\mathcal{H}(G_{n,i},F_i) \to 0\) as \(n \to \infty\). Since, by 2, \(y_{n,i} \in Z_{Y_i}(F_i)\), it follows from the assumption that there exists \(w_{n,i} \in Z_{Y_i}(G_{n,i})\) for every \(n \in \mathbb{N}\) such that \(\|y_{n,i}-w_{n,i}\| \to 0\) as \(n \to \infty\). Note that for every \(n \in \mathbb{N}\), by 2, \(w_n \in Z_{Y}(G_n)\), where \(w_n = (w_{n,i})_{i \in \mathbb{N}}.\) Now, it is enough to prove that \(\|y_n-w_n\| \to 0\). For this, let \(\epsilon>0\). Choose \(j_0 \in \mathbb{N}\) such that,
\(\sum_{i>j_0}\|y_{n,i}\|^p < \epsilon ^p\) for all \(n \in \mathbb{N}\) (by 1);
\(\sum_{i> j_0}rad_{Y_i}(F_i)^p< \frac{\epsilon^p}{2^{p+2}}\) (by 1) and
\(\sum_{i> j_0}\|z_i\|^p < \left( \frac{\epsilon}{4} \right)^p\), for some fixed \(z=(z_i)_{i \in \mathbb{N}} \in F\)
holds. Since \(\mathcal{H}(G_n, F) \to 0\) and \(\mathcal{H}(G_{n,i},F_i) \to 0\) for all \(i \in \mathbb{N}\), choose \(n_0 \in \mathbb{N}\) such that for all \(n \geq n_0,\) we have
\(|rad_Y(G_n)^p- rad_Y(F)^p| < \frac{\epsilon^p}{2^{p+2}}\); \((*)\)
\(\mathcal{H}(G_n,F) < \frac{\epsilon}{8}\); \((**)\)
\(\sum_{i \leq j_0}| rad_{Y_i}(G_{n,i})^p-rad_{Y_i}(F_i)^p|< \frac{\epsilon^p}{2^{p+2}}\) and
\(\sum_{i \leq j_0}\|y_{n,i}-w_{n,i}\|^p < (2\epsilon)^p\).
Let \(n \geq n_0.\) Therefore, by \((*)\) and using 1, \[\begin{align} \sum_{i> j_0}rad_{Y_i}(G_{n,i})^p &< \sum_{i \leq j_0} rad_{Y_i}(F_i)^p- \sum_{i \leq j_0} rad_{Y_i}(G_{n,i})^p+ \sum_{i > j_0} rad_{Y_i}(F_i)^p+ \frac{\epsilon^p}{2^{p+2}}\\ & < \frac{\epsilon^p}{2^{p+2}}+ \frac{\epsilon^p}{2^{p+2}}+ \frac{\epsilon^p}{2^{p+2}}\\ & < \left(\frac{\epsilon}{2}\right)^p. \end{align}\] Since, by \((**)\), \(F \subseteq G_n+ \frac{\epsilon}{8}B_X\) holds, choose \(u_n=(u_{n,i})_{i \in \mathbb{N}} \in G_n\) such that \(\|u_n-z\| \leq \frac{\epsilon}{8}\). Thus, \[\left(\sum_{i> j_0} \|u_{n,i}\|^p\right)^{\frac{1}{p}} \leq \left(\sum_{i> j_0} \|u_{n,i}-z_i\|^p\right)^{\frac{1}{p}}+\left(\sum_{i> j_0} \|z_{i}\|^p\right)^{\frac{1}{p}} < \|u_n-z\| + \frac{\epsilon}{4} < \frac{\epsilon}{2}.\] Therefore, it follows that \[\begin{align} \left(\sum_{i>j_0}\|w_{n,i}\|^p \right)^{\frac{1}{p}} &\leq \left(\sum_{i> j_0} r(w_{n,i},G_{n,i})^p \right)^{\frac{1}{p}}+ \left(\sum_{i>j_0}\|u_{n,i}\|^p \right)^{\frac{1}{p}}\\ & = \left(\sum_{i> j_0} rad_{Y_i}(G_{n,i})^p \right)^{\frac{1}{p}}+ \left(\sum_{i>j_0}\|u_{n,i}\|^p \right)^{\frac{1}{p}}\\ & < \frac{\epsilon}{2}+ \frac{\epsilon}{2}\\ & = \epsilon. \end{align}\] Thus, \[\begin{align} \|y_n-w_n\|^p &= \sum_{i \leq j_0}\|y_{n,i}-w_{n,i}\|^p+ \sum_{i> j_0}\|y_{n,i}-w_{n,i}\|^p\\ & \leq \sum_{i \leq j_0}\|y_{n,i}-w_{n,i}\|^p+ \left(\left(\sum_{i> j_0}\|y_{n,i}\|^p\right)^{\frac{1}{p}} + \left(\sum_{i> j_0}\|w_{n,i}\|^p\right)^{\frac{1}{p}}\right)^p\\ & < (2\epsilon)^{p}+(2\epsilon)^{p}. \end{align}\]
Hence, \(\|y_n-w_n\| \to 0.\) This completes the proof of necessary part.
To prove the converse, let \(j \in \mathbb{N}\) and \(F_j \in CB(X_j).\) Consider \(F= \Pi_{i \in \mathbb{N}} F_i \in \mathcal{P}(X)\), where \(F_i=\{0\}\) for all \(i \in \mathbb{N} {\setminus} \{j\}.\) Since \(Z_Y\) is lHsc on \(\mathcal{P}(X),\) there exists \(\delta>0\) such that \(Z_Y(F) \subseteq Z_Y(E) + \epsilon B_X\), for every \(E \in \mathcal{P}(X)\) with \(\mathcal{H}(F,E) < \delta.\) Let \(G_j \in CB(X_j)\) such that \(\mathcal{H}(F_j,G_j) < \delta.\) Now, we need to show that \(Z_{Y_j}(F_j) \subseteq Z_{Y_j}(G_j)+ \epsilon B_{X_j}\). Consider \(G=\Pi_{i \in \mathbb{N}} G_i \in \mathcal{P}(X)\), where \(G_i=\{0\}\) for every \(i \in \mathbb{N} {\setminus} \{j\}.\) Since \(\mathcal{H}(F,G)= \mathcal{H}(F_j,G_j)\), by assumption, \(Z_Y(F) \subseteq Z_Y(G)+ \epsilon B_X.\) Further, by 2, \(Z_{Y_j}(F_j) \subseteq Z_{Y_j}(G_j)+ \epsilon B_{X_j}.\) Thus, \(Z_{Y_j}\) is lHsc on \(CB(X_j)\). Hence the proof. ◻
The following result on the stability of the continuity properties of the metric projection map is an immediate consequence of the preceding theorem.
Corollary 8. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a proximinal subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\). Then \(P_{Y_i}\) is lHsc (respectively, uHsc) on \(X_i\) for every \(i\in \mathbb{N}\) if and only if \(P_Y\) is lHsc (respectively, uHsc) on \(X\).
From [thm:pds_P1,thrm: P_1 lP_2 lHsc,thrm: lHsc P_1], it is easy to verify the stability of property-\((lP_2),\) as shown in the result that follows. Further, as an outcome, we obtain the stability of locally uniform strong proximinality.
Theorem 13. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}}\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}\). Then \((Y_i, CB(X_i))\) has property-\((lP_2)\) for every \(i\in \mathbb{N}\) if and only if \((Y, \mathcal{P}(X))\) has property-\((lP_2).\)
Corollary 9. Let \(\{X_i: {i\in \mathbb{N}}\}\) be a collection of Banach spaces, \(Y_i\) be a subspace of \(X_i\) for every \(i\in \mathbb{N}\) and \(1\leq p < \infty.\) Let \(X=(\oplus_pX_i)_{i \in \mathbb{N}},\) and \(Y=(\oplus_pY_i)_{i \in \mathbb{N}}.\) Then \(Y_i\) is locally uniformly strongly proximinal on \(X_i\) for every \(i\in \mathbb{N}\) if and only if \(Y\) is locally uniformly strongly proximinal on \(X\).
Funding: No funding was received for conducting this study.
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Conflict of interest: The authors have no conflict of interest to declare.