# A study on various generalizations of Generalized centers $\bm{(GC)}$ in
Banach spaces

Syamantak Das  and  Tanmoy Paul  Dept. of Mathematics
Indian Institute of Technology Hyderabad
India<EMAIL_ADDRESS>&<EMAIL_ADDRESS>
###### Abstract.

In [Generalized centers of finite sets in Banach spaces, Acta Math. Univ.
Comenian. (N.S.) 66(1) (1997), 83–115], Veselý developed the idea of
generalized centers for finite sets in Banach spaces. In this work, we explore
the concept of restricted $\mathscr{F}$-center property for a triplet
$(X,Y,\mathcal{F}(X))$, where $Y$ is a subspace of a Banach space $X$ and
$\mathcal{F}(X)$ is the family of finite subsets of $X$. In addition, we
generalize the analysis to include all closed, bounded subsets of $X$. Similar
to how Lindenstrauss characterized $n.2.I.P.$, we characterize $n.X.I.P.$. So,
it is possible to figure out that $Y$ has $n.X.I.P.$ in $X$ for all natural
numbers $n$ if and only if $\textrm{rad}_{Y}(F)=\textrm{rad}_{X}(F)$ for all
finite subsets $F$ of $Y$. It then turns out that, for all continuous monotone
functions $f$, the $f$-radii viz.
$\textrm{rad}_{Y}^{f}(F),\textrm{rad}_{X}^{f}(F)$ are same whenever the
generalised radii viz. $\textrm{rad}_{Y}(F),\textrm{rad}_{X}(F)$ are same, for
all finite subsets $F$ of $Y$. We establish a variety of characterizations of
central subspaces of Banach spaces. With reference to an appropriate subfamily
of closed and bounded subsets, it appears that a number of function spaces and
subspaces exhibit the restricted weighted Chebyshev center property.

###### Key words and phrases:

Chebyshev center, generalized center, weighted Chebyshev center, central
subspace, $n.X.I.P$.

###### 2000 Mathematics Subject Classification:

Primary 41A28, 41A65 Secondary 46B20 41A50 August 30, 2024

##  1\. Introduction

###  1.1. Objectives:

In this study, we focus on the minimization problem $\inf_{y\in
Y}f(\|z_{1}-y\|,\ldots,\|z_{n}-y\|)$ within the framework of Banach spaces $X$
and its closed subspaces $Y$. The problem involves finding the minimum value
of a given function $f:\mathbb{R}^{n}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$,
which is continuous, monotone (point wise), coercive, corresponding to a
finite subset $\\{z_{1},z_{2},\ldots,z_{n}\\}$ of $X$. Additionally, we
explore variations of this problem in Banach spaces. The result presented in
[18, Theorem 2.7] establishes an association between this phenomenon and the
characteristics of balls at their intersection properties in Banach spaces. We
extend the concept of $f$-centers, as introduced in the work by Veselý in
[18], to encompass all closed and bounded subsets of the space $X$.

In [18], the author observed that the generalized radius for a finite set in a
Banach space remains same when it is considered in the bidual of the space. In
this paper we investigate on the following problem.

######  Problem 1.1.

Let $Y$ be a subspace of $X$. For every finite subset $F$ of $Y$, under what
necessary and sufficient conditions do the generalized radii of $F$ remain the
same when it is viewed as a subset of both $X$ and $Y$?

Additionally, we aim to study various properties viz. $\mathscr{F}$-rcp, wrcp,
$\mathcal{A}-C$-subspace, $(GC)$, $\mathcal{A}$-IP (see Definition 1.3, 1.5)
in various function spaces and their subspaces.

###  1.2. Prerequisites:

We list some standard notations used in this study: $X$ denotes a Banach
space, and by subspace, we indicate a closed linear subspace. $B_{X}$ and
$S_{X}$ represent the closed unit ball and the unit sphere of $X$,
respectively. For $x\in X$ and $r>0$, $B_{X}(x,r)$ denotes the closed ball in
$X$, centered at $x$ and radius $r$. When there is no chance of confusion, we
simply write $B(x,r)$ for the closed ball in $X$. We consider an element $x\in
X$ to be canonically embedded in $X^{**}$.
$\mathcal{F}(X),\mathcal{C}(X),\mathcal{K}(X),\mathcal{WC}(X),\mathcal{B}(X)$,
and $\mathcal{P}(X)$ represent the set of all nonempty finite, closed and
convex, compact, weakly compact, closed and bounded, and power set of $X$,
respectively. It is assumed that the real line corresponds to the scalar field
for the spaces. For a nonempty set $S$, $|S|$ denotes the ordinal that
satisfies $\textrm{card}(S)=\textrm{card}(|S|)$.

For an ordinal $\alpha$, we consider coordinate-wise ordering on a subclass of
$\mathbb{R}_{\geq 0}^{\alpha}$, viz.
$\Gamma(\alpha)=\\{\varphi:\alpha\rightarrow\mathbb{R}_{\geq
0}:\sup_{t\in\alpha}\varphi(t)<\infty\\}$, defined as: for
$\varphi_{1},\varphi_{2}\in\Gamma(\alpha)$, $\varphi_{1}\leq\varphi_{2}$ if
and only if $\varphi_{1}(t)\leq\varphi_{2}(t)$ for all $t\in\alpha$. For an
ordinal $\alpha$, we consider $\Gamma(\alpha)$ to be endowed with the supremum
norm.

######  Definition 1.2.

For an ordinal $\alpha$ and a function
$f:\Gamma(\alpha)\rightarrow\mathbb{R}_{\geq 0}$, we call

  * $(a)$

$f$ is monotone if for $\varphi_{1},\varphi_{2}\in\Gamma(\alpha)$,
$\varphi_{1}\leq\varphi_{2}$ implies $f(\varphi_{1})\leq f(\varphi_{2})$.

  * $(b)$

$f$ is coercive if $f(\varphi)\rightarrow\infty$ as
$\|\varphi\|_{\infty}\rightarrow\infty$.

######  Definition 1.3.

  * $(a)$

Let $Y$ be a subspace of $X$, $F\in\mathcal{B}(X)$, and a function
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$. We define
$r_{f}(x,F)=f(\left(\|x-a\|\right)_{a\in F})$ and
$\textrm{rad}_{Y}^{f}(F)=\inf_{y\in Y}r_{f}(y,F)$ for $x\in X$. The collection
of all points in $Y$ where the infimum that defines $\textrm{rad}_{Y}^{f}(F)$
is achieving is referred to as the restricted $f$-centers of $F$ in $Y$ and is
denoted by $\textrm{Cent}^{f}_{Y}(F)$. That is,
$\textrm{Cent}^{f}_{Y}(F)=\\{y\in Y:r_{f}(y,F)=\textrm{rad}_{Y}^{f}(F)\\}$.
When $Y=X$, then the restricted $f$-centers are called the $f$-centers of $F$
in $X$. In this case, we call $X(Y)$ admits (restricted) $f$-centers for $F$.

  * $(b)$

When $f$ is of the form $f(a)=\sup_{t\in|F|}\rho(t)a(t)$, where
$\rho=(\rho(t))_{t\in|F|}\in\Gamma(|F|)$, then
$r_{f}(x,F),\textrm{rad}_{Y}^{f}(F)$, and $\textrm{Cent}_{Y}^{f}(F)$ are
rewritten as $r_{\rho}(x,F),\textrm{rad}_{Y}^{\rho}(F)$, and
$\textrm{Cent}_{Y}^{\rho}(F)$, respectively. In this case, we refer to the
$f$-centers as the restricted weighted Chebyshev centers. In this case, we
call $Y$ admits weighted Chebyshev center for $F$ for weights $(\rho(t))_{t\in
F}$. When $\rho(t)=1$ for all $t$, we denote the quantities by
$r(x,F),\textrm{rad}_{Y}(F)$ and $\textrm{Cent}_{Y}(F)$ respectively.

  * $(c)$

For a subspace $Y$ of $X$, $\mathfrak{F}\subseteq\mathcal{B}(X)$ and a family
of functions $\mathscr{F}$ consisting of
$f:\Gamma({|F|})\rightarrow\mathbb{R}_{\geq 0}$ for all $F\in\mathfrak{F}$,
the triplet $(X,Y,\mathfrak{F})$ is said to have restricted
$\mathscr{F}$-center property ( $\mathscr{F}$-rcp in short) if for all
$F\in\mathfrak{F}$ and $f\in\mathscr{F}$, we have
$\textrm{Cent}_{Y}^{f}(F)\neq\emptyset$.

  * $(d)$

For a subspace $Y$ of $X$ and $\mathfrak{F}\subseteq\mathcal{B}(X)$, the
triplet $(X,Y,\mathfrak{F})$ said to have restricted weighted Chebyshev center
property (wrcp in short) if for all $F\in\mathfrak{F}$ and bounded weights
$\rho:F\rightarrow\mathbb{R}_{\geq 0}$,
$\textrm{Cent}_{Y}^{\rho}(F)\neq\emptyset$.

The following observation is derived in [18, Theorem 2.7].

######  Theorem 1.4.

For a Banach space $X$ and $\\{a_{1},\cdots,a_{n}\\}\subseteq X$, the
following are equivalent:

  * $(a)$

If $r_{1},\ldots,r_{n}>0$ then
$\cap_{i=1}^{n}B_{X^{**}}(a_{i},r_{i})\neq\emptyset$ implies
$\cap_{i=1}^{n}B_{X}(a_{i},r_{i})\neq\emptyset$.

  * $(b)$

$X$ admits weighted Chebyshev centers for $\\{a_{1},\cdots,a_{n}\\}$ for all
weights $\rho_{1},\cdots,\rho_{n}>0$.

  * $(c)$

$X$ admits $f$-centers for $\\{a_{1},\cdots,a_{n}\\}$ for each continuous
monotone coercive function $f$ on $\mathbb{R}^{n}_{\geq 0}$.

A Banach space $X$ belongs to the class $(GC)$ if it satisfies any one of the
equivalent conditions stated in Theorem 1.4. It is denoted by $X\in(GC)$.

Bandyopadhyay and Rao in [3] introduced the notion of central subspace, which
is a generalization of class $(GC)$ given in [18]. Bandyopadhyay and Dutta in
[2] further generalized the notion of central subspace to
$\mathcal{A}-C$-subspace, for a subfamily $\mathcal{A}$ of $\mathcal{B}(X)$.

######  Definition 1.5.

[2] Let $Y$ be a subspace of Banach space $X$ and $\mathcal{A}$ be a family of
subsets of $Y$.

  * $(a)$

$Y$ is said to have an almost $\mathcal{A}-C$-subspace of $X$ if for all $x\in
X$, $A\in\mathcal{A}$ and $\varepsilon>0$, there exists $y\in Y$ such that
$\|y-a\|\leq\|x-a\|+\varepsilon$ for all $a\in A$.

  * $(b)$

Y is said to be a $\mathcal{A}-C$-subspace of $X$ if $\varepsilon=0$ in $(a)$.

  * $(c)$

If $\mathcal{A}$ is a family of subsets of $X$, then it is referred to that
$X$ has (almost) $\mathcal{A}$-IP, if $X$ is an (almost)
$\mathcal{A}-C$-subspace of $X^{**}$.

Note, $X\in(GC)$ if it has $\mathcal{F}(X)$-IP. Moreover,
$\mathcal{F}(Y)-C$-subspaces are called central subspaces in [3].

######  Definition 1.6.

  * $(a)$

[10] A subspace $Y$ of $X$ is said to have $n.X.I.P.$ in $X$ if for any $n$
closed balls $\\{B_{X}(a_{i},r_{i})\\}_{i=1}^{n}$ having centers in $Y$ and
$\cap_{i=1}^{n}B_{X}(a_{i},r_{i})\neq\emptyset$, then
$\cap_{i=1}^{n}B_{Y}(a_{i},r_{i}+\varepsilon)\neq\emptyset$ for all
$\varepsilon>0$.

  * $(b)$

[10] A subspace $Y$ of $X$ is said to be an ideal of $X$ if there exists a
projection $P:X^{*}\rightarrow X^{*}$ such that $\ker(P)=Y^{\perp}$.

Thus, the subspaces $Y$ those have $n.X.I.P.$ for all $n$ are precisely those
which are almost $\mathcal{F}(Y)-C$-subspaces of $X$. In general, $n.X.I.P.$
does not imply $k.X.I.P.$ for $k>n$, although under certain assumptions,
$n.X.I.P.$ for all $n$ is equivalent to $3.X.I.P.$.

A subspace $Y$ of $X$ which is known to be an ideal in $X$ also satisfies
$n.X.I.P.$ (see [10, Proposition 3.2]). A subspace that is a range of a
norm-$1$ projection is clearly a central subspace and also an ideal. Let $Y$
be a subspace of $X$, we introduce the following notions here.

######  Definition 1.7.

A subspace $Y$ of $X$ is said to have restricted $n.X.I.P.$ in $X$ (restricted
central subspace of $X$) if for any $n$ closed balls
$\\{B_{X}(a_{i},r)\\}_{i=1}^{n}$ having centers in $Y$ and
$\cap_{i=1}^{n}B_{X}(a_{i},r)\neq\emptyset$, then
$\cap_{i=1}^{n}B_{Y}(a_{i},r+\varepsilon)\neq\emptyset$ for all
$\varepsilon>0$ ($\cap_{i=1}^{n}B_{Y}(a_{i},r)\neq\emptyset$).

For a Banach space $X$, and a finite measure space $(\Omega,\Sigma,\mu)$ and
$1\leq p<\infty$, $L_{p}(\Sigma,X)$ represents the space of all $p$th-Bochner
integrable functions, which are precisely $f:\Omega\to X$ measurable and
$\int_{\Omega}\|f(t)\|^{p}d\mu(t)<\infty$.
$\|f\|_{p}:=\left(\int_{\Omega}\|f(t)\|^{p}d\mu(t)\right)^{1/p}$ defines a
norm on $L_{p}(\Sigma,X)$, which makes $L_{p}(\Sigma,X)$ a Banach space.
$L_{\infty}(\Sigma,X)$ represents the set of all essentially bounded functions
$f:\Omega\rightarrow X$, which are measurable. [6, Ch. 2] is a standard
reference for these spaces and all the properties used in this article.

Let us recall from [6, Ch. 5] that for a sub $\sigma$-algebra
$\Sigma^{\prime}\subseteq\Sigma$, by considering $L_{p}(\Sigma^{\prime},X)$ as
a subspace of $L_{p}(\Sigma,X)$, $1\leq p<\infty$, the conditional expectation
operator is a mapping $E:L_{p}(\Sigma,X)\rightarrow L_{p}(\Sigma^{\prime},X)$
such that $E(f)=g$, where $\int_{B}fd\mu=\int_{B}gd\mu$ for all
$B\in\Sigma^{\prime}$ indicates a linear projection of norm-$1$.

######  Definition 1.8.

[7]

  * $(a)$

A linear projection $P:X\rightarrow X$ is said to be an $L$-projection if
$\|x\|=\|Px\|+\|x-Px\|$ for all $x\in X$.

  * $(b)$

A Banach space $X$ is said to be $L$-embedded if $X$, under its canonical
image in $X^{**}$, is the range of an $L$-projection on $X^{**}$.

  * $(c)$

A closed subspace $J\subseteq X$ is an $M$-ideal in $X$ if $J^{\perp}$ is the
range of an $L$-projection in $X^{*}$([7]).

Reference [7, Ch. 4] provides examples and other properties of $L$-embedded
spaces. We call a subspace $Y$ of $X$ is $1$-complemented if $Y$ is a range of
a norm-$1$ projection from $X$. If $Y$ is a $1$-complemented subspace of an
$L$-embedded space $X$, then $Y$ is also $L$-embedded (see [7, Theorem
IV.1.5.]). For a sub $\sigma$-algebra $\Sigma^{\prime}\subseteq\Sigma$, due to
the conditional expectation $E:L_{1}(\Sigma,X)\rightarrow
L_{1}(\Sigma^{\prime},X)$, $L_{1}(\Sigma^{\prime},X)$ is $L$-embedded if
$L_{1}(\Sigma,X)$ is so.

A Banach space $X$ is said to be a Lindenstrauss space (or $L_{1}$-predual) if
$X^{*}\cong L_{1}(\mu)$ for some measure $\mu$. In [11], Lindenstrauss
characterizes these spaces as Banach spaces, where any collection of pairwise
intersecting closed balls whose centers form a compact set has a nonempty
intersection.

An ideal of a Lindenstrauss space is also a central subspace (see [9, Lemma
10, Theorem 15]). In [18], Veselý derived that for spaces $X$ with the Radon-
Nikodým-property (RNP in short) and $1$-complemented in its bidual,
$L_{p}(\mu,X)\in(GC)$, for $1\leq p<\infty$. Additionally, if $X$ is a dual
space that is strictly convex and has $(w^{*}K)$, then $C_{b}(T,X)\in(GC)$ for
a Hausdorff space $T$.

We refer to the articles [2, 18] for various examples of the spaces discussed
in this paper.

###  1.3. Observations:

In this subsection, we give overview of our observations. Suppose that, $Y$
has $n.X.I.P.$ in $X$ and $F\in\mathcal{F}(Y)$, where $|F|=n$ and $X$ admits
weighted Chebyshev centers for $F$. In section 2, it is demonstrated that $Y$
admits restricted $f$-centers for $F$, where $|F|=n$, for all continuous,
monotone functions $f:\mathbb{R}^{n}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$,
if and only if for any collection of $n$ closed balls with centers in $F$
having a nonempty intersection in $X$ also intersect in $Y$. We obtain a
similar characterization to that [18, Theorem 2.7] for the family of bounded
subsets of $Y$.

Problem 1.1 is answered in Theorem 2.7, which extends a characterization to
the notion $n.X.I.P.$ in Banach spaces. Consequently, we obtain various
characterizations for central subspaces in Banach spaces, as stated in Theorem
2.8 and Theorem 2.9. It is easy to observe that if $Y$ is a subspace of $X$
and if there exists $P:X\rightarrow Y$ onto, where $P(\lambda x+y)=\lambda
P(x)+y$, for $x\in X,y\in Y$ and scalar $\lambda$, $\|P\|\leq 1$, then for any
$F\in\mathcal{B}(Y)$, $\textrm{rad}_{Y}(F)=\textrm{rad}_{X}(F)$. Veselý’s
example in [18, Pg.9] ensures there does not exist any quasi linear projection
$P:\ell_{\infty}\rightarrow c_{0}$. We do not know the solution to the
question posed in Problem 1.1 for closed bounded subsets, with the exception
of some trivial circumstances, such as when $Y$ is $1$-complemented in $X$ or
more generally a range of a quasi linear projection. The requirement for $Y$
being a central subspace can be more simply expressed when $Y\in(GC)$ is
observed.

Let us give an example of a space $X$ such that $L_{1}(\mu,X)\notin(GC)$. Let
$f\in\ell_{1}$ be such that support of $f$ is infinite and
$2\|f\|_{\infty}>\|f\|_{1}$. Now identify $f$ as a linear functional on
$c_{0}$ and suppose that $X=\ker(f)$. Thus, $X\notin(GC)$ (see [17]). Since
$X$ is $1$-complemented in $L_{1}(\mu,X)$, $L_{1}(\mu,X)\notin(GC)$.

In section 3, we investigate the restricted $\mathscr{F}$-center property for
various triplets $(X,Y,\mathfrak{F})$, where $Y$ represents a subspace of $X$
and $\mathfrak{F}\subseteq\mathcal{B}(X)$. It is evident that when a given
subspace $Y\subseteq X$, $(X,Y,\mathcal{F}(X))$ possesses wrcp, then
$Y\in(GC)$. On the other hand, from Theorem 2.3 one can conclude in the above
instances $Y$ is a central subspace of $X$ whenever $X$ admits $f$-centers for
$F\in\mathcal{F}(Y)$, for all continuous monotone functions
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$.

Methods for dealing with the aforementioned ideas, including $(GC)$, central
subspace, $n.X.I.P.$, $\mathcal{A}-C$-subspace are adapted from [2, 3, 16, 17,
18]. In section 3, we use certain measure-theoretic tools from [6] to derive
our observations.

##  2\. $\mathcal{A}-C$-subspaces of Banach spaces

The following is obtained by employing justifications similar to those given
in [18, Theorem 2.6].

######  Theorem 2.1.

Let $Y$ be a subspace of $X$ and $n\in\mathbb{N}$. Suppose $Y$ has $n.X.I.P.$
in $X$, and $f:\mathbb{R}^{n}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$ be a
continuous, monotone function. Then, for all $F\in\mathcal{F}(Y)$ with
$|F|=n$, we have $\emph{rad}_{Y}^{f}(F)=\emph{rad}_{X}^{f}(F)$.

###### Proof.

Choose $\varepsilon>0$ and let $x\in X$ be such that,
$f(\|x-z_{1}\|,\ldots,\|x-z_{n}\|)<\textrm{rad}_{X}^{f}(F)+\varepsilon$. Since
$Y$ has $n.X.I.P.$ in $X$, there exists
$y\in\cap_{i=1}^{n}B_{Y}(a_{i},\|x-a_{i}\|+\varepsilon)$.

$\displaystyle\mbox{Now\leavevmode\nobreak\ }\textrm{rad}_{Y}^{f}(F)$
$\displaystyle\leq$ $\displaystyle f(\|y-z_{1}\|,\ldots,\|y-z_{n}\|)$
$\displaystyle\leq$ $\displaystyle
f(\|x-z_{1}\|+\varepsilon,\ldots,\|x-z_{n}\|+\varepsilon)$ $\displaystyle\leq$
$\displaystyle
f(\|x-z_{1}\|,\ldots,\|x-z_{n}\|)+\varepsilon\leq\textrm{rad}_{X}^{f}(F)+2\varepsilon.$

Since $\varepsilon>0$ is arbitrary the result follows. ∎

The proof of the following theorem follows in the same manner as the proof of
[18, Theorem 2.7].

######  Theorem 2.2.

Assume that $F\in\mathcal{F}(Y)$, where $F=\\{y_{1},\cdots,y_{n}\\}$. Suppose
that for each continuous, monotone function $f:\mathbb{R}^{n}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$,
$\emph{rad}_{Y}^{f}(F)=\emph{rad}_{X}^{f}(F)$ and $X$ admits $f$-centers for
$F$. Then the following are equivalent.

  * $(a)$

If for $r_{1},\cdots,r_{n}>0$,
$\cap_{i=1}^{n}B_{X}(y_{i},r_{i})\neq\emptyset$, then
$\cap_{i=1}^{n}B_{Y}(y_{i},r_{i})\neq\emptyset$.

  * $(b)$

$Y$ admits restricted weighted Chebyshev centers for $F$ for all weights
$\rho_{1},\cdots,\rho_{n}>0$.

  * $(c)$

$Y$ admits restricted $f$-centers for $F$ for all continuous, monotone
functions $f:\mathbb{R}^{n}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$.

###### Proof.

$(a)\Rightarrow(c)$. Let $f:\mathbb{R}^{n}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}$ be a continuous, monotone function and $x\in\textrm{Cent}_{X}^{f}(F)$. As
$x\in\cap_{i=1}^{n}B_{X}(y_{i},\|x-y_{i}\|)$, there exists
$y\in\cap_{i=1}^{n}B_{Y}(y_{i},\|x-y_{i}\|)$. Since
$\textrm{rad}_{Y}^{f}(F)=\textrm{rad}_{X}^{f}(F)$, it can be clearly
understood that $y\in\textrm{Cent}_{Y}^{f}(F)$.

$(c)\Rightarrow(b)$. This is obvious.

$(b)\Rightarrow(a)$. Let $x\in\cap_{i=1}^{n}B_{X}(y_{i},r_{i})$. Define
$\rho_{i}=\frac{1}{r_{i}}$ for all $i=1,\cdots,n$. Accordingly,
$r_{\rho}(x,F)\leq 1$. Based on our assumption, we obtain
$\textrm{rad}_{Y}^{\rho}(F)=\textrm{rad}_{X}^{\rho}(F)\leq r_{\rho}(x,F)\leq
1$. Moreover, there exists $y\in Y$ such that
$y\in\textrm{Cent}_{Y}^{\rho}(F)$. Thus $\frac{1}{r_{i}}\|y-y_{i}\|\leq 1$ for
all $i=1,\cdots,n$. Thus $y\in\cap_{i=1}^{n}B_{Y}(y_{i}.r_{i})$. ∎

By varying $F\in\mathcal{F}(Y)$ in Theorem 2.2 we obtain the following.

######  Theorem 2.3.

Let $Y$ be a subspace of $X$ and for all $F\in\mathcal{F}(Y)$,
$\emph{rad}_{Y}^{f}(F)=\emph{rad}_{X}^{f}(F)$ and $X$ admits $f$-centers for
$F$, for all continuous monotone $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq
0}$. Then the following are equivalent.

  * $(a)$

$Y$ is a central subspace of $X$.

  * $(b)$

The triplet $(X,Y,\mathcal{F}(Y))$ has wrcp.

  * $(c)$

$\emph{Cent}_{Y}^{f}(F)=\emph{ Cent}_{X}^{f}(F)\cap Y\neq\emptyset$, for all
$F\in\mathcal{F}(Y)$.

######  Remark 2.4.

For the bounded subsets of $Y$, Theorem 2.3 also holds true.

######  Lemma 2.5.

Let $Y$ be a subspace of $X$. If $Y$ has restricted $n.X.I.P.$ in $X$, then
$Y$ has $n.X.I.P.$ in $X$.

###### Proof.

Let $\\{B_{X}(y_{i},r_{i})\\}_{i=1}^{n}$ be a collection of $n$ closed balls
in $X$, where $y_{1},\cdots,y_{n}\in Y$ and $r_{1},\cdots,r_{n}>0$ such that
$\cap_{i=1}^{n}B_{X}(y_{i},r_{i})\neq\emptyset$.

Suppose there exists an $\varepsilon>0$ for which
$\cap_{i=1}^{n}B_{Y}(y_{i},r_{i}+\varepsilon)=\emptyset$. Let $r\in\mathbb{R}$
be such that $r-\frac{\varepsilon}{2}>r_{i}$ for all $i=1,\cdots,n$. We now
construct $n$ closed balls
$\\{B_{X}(w_{i},r-\frac{\varepsilon}{2})\\}_{i=1}^{n}$ with
$w_{1}\cdots,w_{n}\in Y$ such that
$B_{X}(w_{i},r-\frac{\varepsilon}{2})\supseteq B_{X}(y_{i},r_{i})$ and
$\cap_{i=1}^{n}B_{Y}(w_{i},r)=\emptyset$, which contradicts our assumption. We
now choose $w_{i}$ inductively. To this end, suppose that we have $w_{i}$ for
$i\leq j(<n)$ such that,

$\displaystyle B_{X}(w_{i},r-\frac{\varepsilon}{2})\supseteq
B_{X}(y_{i},r_{i})\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ i\leq
j\leavevmode\nobreak\ \text{and}$
$\displaystyle\cap_{i=1}^{j}B_{Y}(w_{i},r)\cap\left(\cap_{i=j+1}^{n}B_{Y}(y_{i},r_{i}+\varepsilon)\right)=\emptyset.$

Let us define
$K_{1}=\cap_{i=1}^{j}B_{Y}(w_{i},r)\cap\left(\cap_{i=j+2}^{n}B_{Y}(y_{i},r_{i}+\varepsilon)\right)$
and $K_{2}=B_{Y}(y_{j+1},r_{j+1}+\varepsilon)$.

Case 1: Suppose $K_{1}=\emptyset$. Accordingly, we choose $w_{j+1}=y_{j+1}$
and obtain
$\cap_{i=1}^{j+1}B_{Y}(w_{i},r)\cap\left(\cap_{i=j+2}^{n}B_{Y}(y_{i},r_{i}+\varepsilon)\right)=\emptyset$
and proceed further to choose $w_{j+2}$.

Case 2: Assume that $K_{1}\neq\emptyset$. Thus, $K_{1}\cap K_{2}=\emptyset$
and so $(K_{1}-y_{j+1})\cap B_{Y}(0,r_{j+1}+\varepsilon)=\emptyset$. Thus, we
obtain $f\in Y^{*}$ such that $f(y-y_{j+1})\geq 1\geq f(x)$ for all $y\in
K_{1}$ and $x\in B_{Y}(0,r_{j+1}+\varepsilon)$.

Now, $\|f\|=\sup_{y\in B_{Y}}f(y)=\frac{1}{r_{j+1}+\varepsilon}\sup_{y\in
B_{Y}(0,r_{j+1}+\varepsilon)}f(y)\leq\frac{1}{r_{j+1}+\varepsilon}$.

Choose $\delta>0$ such that
$\delta<\frac{1}{r-\frac{\varepsilon}{2}-r_{j+1}}\big{(}1-\frac{r_{j+1}+\frac{\varepsilon}{2}}{r_{j+1}+\varepsilon}\big{)}$.
Let $z\in S_{Y}$ such that $f(z)\leq-\|f\|+\delta$. Let
$w_{j+1}=y_{j+1}+(r-\frac{\varepsilon}{2}-r_{j+1})z\in Y$.

Let $x\in B_{X}(y_{j+1},r_{j+1})$. Thus,

$\displaystyle\|x-w_{j+1}\|$ $\displaystyle\leq$
$\displaystyle\|x-y_{j+1}\|+(r-\frac{\varepsilon}{2}-r_{j+1})\|z\|$
$\displaystyle\leq$ $\displaystyle r_{j+1}+(r-\frac{\varepsilon}{2}-r_{j+1})$
$\displaystyle=$ $\displaystyle r-\frac{\varepsilon}{2}.$

Thus, $x\in B_{X}(w_{j+1},r-\frac{\varepsilon}{2})$ and hence
$B_{X}(y_{j+1},r_{j+1})\subseteq B_{X}(w_{j+1},r-\frac{\varepsilon}{2})$.

Let $y\in B_{Y}(w_{j+1},r)$. Thus,

$\displaystyle f(y-y_{j+1})$ $\displaystyle=$ $\displaystyle
f(y-w_{j+1})+f(w_{j+1}-y_{j+1})$ $\displaystyle\leq$
$\displaystyle\|f\|r+(r-\frac{\varepsilon}{2}-r_{j+1})f(z)$
$\displaystyle\leq$
$\displaystyle\|f\|r+(r-\frac{\varepsilon}{2}-r_{j+1})(-\|f\|+\delta)$
$\displaystyle\leq$
$\displaystyle\frac{r_{j+1}+\frac{\varepsilon}{2}}{r_{j+1}+\varepsilon}+\delta(r-\frac{\varepsilon}{2}-r_{j+1})<1.$

Thus, for all $y\in B_{Y}(w_{j+1},r)\cap Y$, $y\notin K_{1}$. Hence,
$K_{1}\cap B_{Y}(w_{j+1},r)=\emptyset$.

This completes the proof. ∎

######  Corollary 2.6.

Let $Y$ be a subspace of $X$ such that $\emph{rad}_{Y}(F)=\emph{rad}_{X}(F)$
for all $F\in\mathcal{F}(Y)$. Then for all $n\in\mathbb{N}$, $Y$ possesses
$n.X.I.P.$ in $X$.

###### Proof.

Let $y_{1},\ldots,y_{n}\in Y$ and $\\{B_{X}(y_{i},r)\\}_{i=1}^{n}$ be a finite
family of closed balls in $X$, where
$\cap_{i=1}^{n}B_{X}(y_{i},r)\neq\emptyset$ in $X$. Hence,
$\textrm{rad}_{Y}(F)=\textrm{rad}_{X}(F)\leq r$. As
$\cap_{i=1}^{n}B_{Y}(y_{i},\textrm{rad}_{Y}(F)+\varepsilon)\neq\emptyset$
holds for all $\varepsilon>0$, we have that the balls
$\\{B_{X}(y_{i},r)\\}_{i=1}^{n}$ almost intersect in $Y$. The result now
follows from Lemma 2.5. ∎

Combining Theorem 2.1, Lemma 2.5, and Corollary 2.6, we obtain the following.

######  Theorem 2.7.

Let $Y$ be a subspace of $X$. Then the following are equivalent.

  * $(a)$

$Y$ has $n.X.I.P.$ in $X$, for all $n$.

  * $(b)$

$Y$ has restricted $n.X.I.P.$ in $X$, for all $n$.

  * $(c)$

$\emph{rad}_{Y}(F)=\emph{rad}_{X}(F)$, for all $F\in\mathcal{F}(Y)$.

  * $(d)$

$\emph{rad}_{Y}^{f}(F)=\emph{rad}_{X}^{f}(F)$, for all $F\in\mathcal{F}(Y)$,
for all continuous, monotone functions $f:\mathbb{R}^{n}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$, $n\geq 1$.

We now derive a few characterizations of central subspaces of Banach spaces.

######  Theorem 2.8.

Let $Y$ be a subspace of $X$ and $Y\in(GC)$. Thus, $Y$ is a restricted central
subspace of $X$ if and only if $Y$ is a central subspace of $X$.

###### Proof.

Let $\\{B_{X}(y_{i},r_{i})\\}_{i=1}^{n}$ be a collection of $n$ closed balls
in $X$, where $y_{1},\cdots,y_{n}\in Y$ and $r_{1},\cdots,r_{n}>0$ such that
$\cap_{i=1}^{n}B_{X}(y_{i},r_{i})\neq\emptyset$.

Claim: $\cap_{i=1}^{n}B_{Y}(y_{i},r_{i})\neq\emptyset$.

By [3, Proposition 2.9], it can be sufficiently shown that
$\cap_{i=1}^{n}B_{Y}(y_{i},r_{i}+\varepsilon)\neq\emptyset$ for all
$\varepsilon>0$. Accordingly, from Lemma 2.5, we conclude the proof. ∎

######  Theorem 2.9.

Let $Y$ be a subspace of $X$ and $X$ admits weighted Chebyshev centers for all
finite subsets of $Y$. Suppose that $Y$ has $n.X.I.P.$ in $X$ for all $n$.
Then, $Y$ is a central subspace of $X$ if and only if $Y\in(GC)$.

###### Proof.

Suppose that $Y$ is a central subspace of $X$. Thus, from Theorem 2.1 and 2.2,
we obtain $(X,Y,\mathcal{F}(Y))$ has wrcp.

Conversely, assume that $(X,Y,\mathcal{F}(Y))$ has wrcp. Since
$(X,Y,\mathcal{F}(Y))$ has wrcp, $Y\in(GC)$. Hence by Theorem 2.8, it is
sufficient to consider balls with centers in $Y$ of equal radii. Let
$\\{B_{X}(y_{i},r)\\}_{i=1}^{n}$ be a collection of $n$ balls with
$y_{1},\cdots,y_{n}\in Y$, $r>0$ such that
$\cap_{i=1}^{n}B_{X}(y_{i},r)\neq\emptyset$. Let $F=\\{y_{1},\cdots,y_{n}\\}$.
Thus, $\textrm{rad}_{X}(F)\leq r$ and by our assumption,
$\textrm{rad}_{Y}(F)\leq r$. Owing to the fact that $(X,Y,\mathcal{F}(Y))$ has
wrcp, $\cap_{i=1}^{n}B_{Y}(y_{i},\textrm{rad}_{Y}(F))\neq\emptyset$ and
consequently $\cap_{i=1}^{n}B_{Y}(y_{i},r)\neq\emptyset$. ∎

######  Theorem 2.10.

Let $\mathcal{A}$ be any of the families
$\mathcal{F},\mathcal{K},\mathcal{B}$, or $\mathcal{P}$, $P$ be a projection
on $X$ of norm-$1$, $Y=\ker(P)$, and $Z\subseteq P(X)$ be a subspace. If $Y+Z$
is an (almost) $\mathcal{A}(Y+Z)-C$-subspace of $X$, then $Z$ is an (almost)
$\mathcal{A}(Z)-C$-subspace of $P(X)$.

###### Proof.

Let $A\in\mathcal{A}(Z)$. Suppose $\underset{a\in A}{\cap}B_{X}(a,r_{a})\cap
P(X)\neq\emptyset$. Thus, $\underset{a\in A}{\cap}B_{X}(a,r_{a})\cap
X\neq\emptyset$. Based on our assumption, $\underset{a\in
A}{\cap}B_{X}(a,r_{a})\cap(Y+Z)\neq\emptyset$. Let
$y_{0}+z_{0}\in\underset{a\in A}{\cap}B_{X}(a,r_{a})\cap(Y+Z)\neq\emptyset$.
Accordingly, for all $a\in A$,

$\displaystyle\|z_{0}-a\|$ $\displaystyle=$
$\displaystyle\|P(z_{0}-y_{0}-a)\|\leq\|z_{0}-y_{0}-a\|\leq r_{a}.$

This proves $Z$ is an $\mathcal{A}(Z)-C$-subspace of $P(X)$. The remaining
follows in a similar way as stated above. ∎

######  Proposition 2.11.

Let $\mathcal{A}$ be any of the families
$\mathcal{F},\mathcal{K},\mathcal{B}$, or $\mathcal{P}$. Let $Z\subseteq
Y\subseteq X$ represent closed subspaces of $X$ such that $Y$ is an M-ideal in
$X$. If $Z$ is an (almost) $\mathcal{A}(Z)-C$-subspace of $Y$ and $Y$ has
(almost) $\mathcal{A}(Y)$-IP, then $Z$ is an (almost)
$\mathcal{A}(Z)-C$-subspace of $X$.

###### Proof.

Let $A\in\mathcal{A}(Z)$ and $\\{B_{X}(a,r_{a})\\}_{a\in A}$ be a collection
of closed balls, where $r_{a}>0$ for all $a\in A$, such that $\cap_{a\in
A}B_{X}(a,r_{a})\neq\emptyset$. Since $Y$ is an $M$-ideal in $X$, we obtain
$X^{**}=Y^{\perp\perp}\oplus_{\infty}N^{\perp}$ for a closed subspace $N$ of
$X^{*}$. If $x\in\cap_{a\in A}B_{X}(a,r_{a})$, then it can be assumed that
$x=y+z$, where $y\in Y^{\perp\perp}$ and $z\in N^{\perp}$. Now
$\max\\{\|y-a\|,\|z\|\\}\leq r_{a}$ for all $a\in A$. Since $Y$ has
$\mathcal{A}(Y)$-IP, there exists $y_{0}\in Y$ such that $\|y_{0}-a\|\leq
r_{a}$ for all $a\in A$. Thus, based on our assumption, $\cap_{a\in
A}B_{Z}(a,r_{a})\neq\emptyset$. ∎

Theorem 2.12 and Corollary 2.14 indicate the consequences of our previous
observations and are derived from the techniques employed in [19, Theorem 2.1,
Corollary 2.2]. We now use the well-known Michael selection theorem ([13,
Theorem 3.2${}^{{}^{\prime\prime}}$]) several times to obtain our
observations.

######  Theorem 2.12.

Let $X$ be a Lindenstrauss space, $K$ and $S$ compact Hausdorff spaces, and
$\psi:K\rightarrow S$ a continuous onto map. Let $\psi^{*}:C(S,X)\rightarrow
C(K,X)$ be a continuous isometric embedding expressed as $\psi^{*}f=f\psi$.
Accordingly, $\psi^{*}C(S,X)$ is a $\mathcal{K}(\psi^{*}C(S,X))-C$-subspace of
$C(K,X)$.

###### Proof.

We denote the space $C(K,X)$ by $\mathfrak{X}$. Let
$A\in\mathcal{K}(\psi^{*}C(S,X))$ and $B=\\{f\in C(S,X):\psi^{*}f\in A\\}$.
Suppose that, $\cap_{f\in B}B_{\mathfrak{X}}(\psi^{*}f,r_{f})\neq\emptyset$,
where $r_{f}>0$ for all $f\in B$. Let us define a multivalued map
$F:S\rightarrow\mathcal{C}(X)$ by $F(y)=\cap_{f\in B}B_{X}(f(y),r_{f})$ is
defined for all $y\in S$. Evidently, $F(y)$ is closed, convex, and
$F(y)\neq\emptyset$ for all $y\in S$.

Claim : $F$ is lower semicontinuous.

Let $G\subseteq X$ be open and $y_{0}\in\\{y\in S:F(y)\cap G\neq\emptyset\\}$.
Let $a\in F(y_{0})\cap G$. Thus, we have $a\in\cap_{f\in
B}B_{X}(f(y_{0}),r_{f})$ and $B_{X}(a,\varepsilon)\subseteq G$ for some
$\varepsilon>0$. Accordingly, there exists an open neighbourhood $N$ of
$y_{0}$ such that $\|f(y)-f(y_{0})\|<\varepsilon$ for all $y\in N$ and $f\in
B$. Now for all $y\in N$, we have $B_{X}(f(y),r_{f})\cap
B_{X}(a,\varepsilon)\neq\emptyset$ for all $f\in B$ and since $X$ is a
Lindenstrauss space, $F(y)\cap G\neq\emptyset$. Thus, $N\subseteq\\{y\in
S:F(y)\cap G\neq\emptyset\\}$ and $F$ is lower semicontinuous. Thus, in
accordance with Michael’s selection theorem, there exists a continuous
$h:S\rightarrow X$ such that $h(y)\in F(y)$ for all $y\in S$. It is evident
that $\psi^{*}h\in\cap_{f\in
B}B_{\mathfrak{X}}(\psi^{*}f,r_{f})\cap\psi^{*}C(S,X)$. ∎

######  Remark 2.13.

Suppose that, the sets $S,X$ and $\psi^{*}$ to be the elements as specified in
Theorem 2.12. Since $\psi^{*}C(S,X)$ is a central subspace of a Lindenstrauss
space, it follows that $\psi^{*}C(S,X)$ is also a Lindenstrauss space.

######  Corollary 2.14.

Let $S,K,\psi$ and $X$ be as in Theorem 2.12. Furthermore, fix $y_{0}\in S$
and let $M=\\{\psi^{*}f:f\in C(S,X)\leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ f(y_{0})=0\\}$. Then, $M$ is a
$\mathcal{K}(M)-C$-subspace of $C(K,X)$.

###### Proof.

We denote the space $C(K,X)$ by $\mathfrak{X}$. Let $A\in\mathcal{K}(M)$ and
$B=\\{f\in C(S,X):\psi^{*}f\in A\\}$. Suppose that, $\cap_{f\in
B}B_{\mathfrak{X}}(\psi^{*}f,r_{f})\neq\emptyset$, where $r_{f}>0$ for all
$f\in B$. Define $F:S\rightarrow\mathcal{C}(X)$ by $F(y)=\cap_{f\in
B}B(f(y),r_{f})$ for all $y\in S$. As in the proof of Theorem 2.12, we obtain
$F(y)\neq\emptyset$ for all $y\in S$. Now we define
$F_{0}:S\rightarrow\mathcal{C}(X)$ by:

$F_{0}(y)=\begin{cases}F(y)\quad&\text{if}\,y\neq y_{0}\\\
\\{0\\}\quad&\text{if}\,y=y_{0}\\\ \end{cases}$

Clearly, $F_{0}$ is closed, convex and $F_{0}(y)\neq\emptyset$ for all $y\in
S$. Similarly, as done in the proof of Theorem 2.12, it can be shown that
$F_{0}$ is lower semicontinuous. Thus, by Michael’s selection theorem, we
obtain $h:S\rightarrow X$ such that $h(y)\in F_{0}(y)$ for all $y\in S$. Thus,
$\psi^{*}h\in\cap_{f\in B}B_{\mathfrak{X}}(\psi^{*}f,r_{f})\cap M$. ∎

Proceeding similarly to the proof of Theorem 2.12, the following can be
obtained.

######  Corollary 2.15.

  * $(a)$

Let $X$ be a Lindenstrauss space, $T$ compact Hausdorff space and $S$ be a
closed subset of $T$. If $M=\\{f\in C(T,X):f|_{S}=0\\}$, then $M$ is a
$\mathcal{K}(M)-C$-subspace of $C(T,X)$.

  * $(b)$

Let $X$ be a Lindenstrauss space and $T$ be a compact Hausdorff space. Then,
$C(T,X)$ is an $\mathcal{K}(C(T,X))-C$-subspace of $\ell_{\infty}(T,X)$.

Our following observation shows that the characteristic of being a central
subspace is stable in the spaces of continuous functions. Let us recall the
manner in which the Lindenstrauss spaces are characterized in the
introduction.

######  Theorem 2.16.

Let $\mathcal{A}$ be any of the families $\mathcal{F},\mathcal{K}$. Let $X$ be
a Lindenstrauss space, $Y$ be a $\mathcal{A}(Y)-C$-subspace of $X$, and $T$ be
a compact Hausdorff space. Then, $C(T,Y)$ is a
$\mathcal{A}(C(T,Y))-C$-subspace of $\ell_{\infty}(T,X)$.

###### Proof.

We denote the space $\ell_{\infty}(T,X)$ by $\mathfrak{X}$. We only prove the
result for $\mathcal{A}=\mathcal{K}$, the case for $\mathcal{A}=\mathcal{F}$
follows by using similar arguments. Let $A\in\mathcal{K}(C(T,Y))$ and
$\cap_{f\in A}B_{\mathfrak{X}}(f,r_{f})\neq\emptyset$, where $r_{f}>0$ for all
$f\in A$. Define $F:T\rightarrow\mathcal{C}(Y)$ by $F(t)=\cap_{f\in
A}B_{Y}(f(t),r_{f})$. Now since for all $t\in T$, $\cap_{f\in
A}B_{X}(f(t),r_{f})\neq\emptyset$, based on our assumption, $\cap_{f\in
A}B_{Y}(f(t),r_{f})\cap Y\neq\emptyset$ for all $t\in T$. Clearly, $F(t)$ is
closed and convex for all $t\in T$.

Claim : $F$ is lower semicontinuous.

Let $G\subseteq X$ be open. Let $t_{0}\in\\{t\in T:F(t)\cap
G\neq\emptyset\\}$. Let $a\in F(t_{0})\cap G$. Thus, $a\in\cap_{f\in
A}B_{Y}(f(t_{0}),r_{f})$ and $B_{X}(a,\varepsilon)\subseteq G$ for some
$\varepsilon>0$. From the continuity of $f$, we obtain an open neighbourhood
$N$ of $t_{0}$ such that $\|f(t)-f(t_{0})\|<\varepsilon$ for all $t\in N$.
Accordingly, $B_{X}(f(t),r_{f})\cap B_{X}(a,\varepsilon)\neq\emptyset$ for all
$f\in A$ and since $X$ is a Lindenstrauss space, we have $\cap_{f\in
A}B(f(t),r_{f})\cap B(a,\varepsilon)\neq\emptyset$ for all $t\in N$. Based on
our assumption, we have $\cap_{f\in A}B_{X}(f(t),r_{f})\cap
B_{X}(a,\varepsilon)\cap Y\neq\emptyset$ for all $t\in N$. Thus,
$N\subseteq\\{t\in T:F(t)\cap G\neq\emptyset\\}$ and so $F$ is lower
semicontinuous. By Michael’s selection theorem, we obtain $h:T\rightarrow Y$
such that $h(t)\in F(t)$ for all $t\in T$. This completes the proof. ∎

We end this section by providing a straightforward application of the
conditional expectation projection $E:L_{p}(\Sigma,X)\rightarrow
L_{p}(\Sigma^{\prime},X)$, for $1\leq p<\infty$.

In the following $(\Omega,\Sigma,\mu)$ denotes a finite measure space and
$\Sigma^{\prime}$ be the sub $\sigma$-algebra of $\Sigma$.

######  Proposition 2.17.

Let $Y$ be a subspace of $X$ and $\mathcal{A}$ be any of the families
$\mathcal{F},\mathcal{K},\mathcal{B}$, or $\mathcal{P}$. Let $L_{1}(\Sigma,Y)$
be a $\mathcal{A}(L_{1}(\Sigma,Y))-C$-subspace of $L_{1}(\Sigma,X)$.
Consequently, $L_{1}(\Sigma^{\prime},Y)$ is an
$\mathcal{A}(L_{1}(\Sigma^{\prime},Y))-C$-subspace of
$L_{1}(\Sigma^{\prime},X)$.

######  Remark 2.18.

In section 3, we encounter cases in Theorem 3.10 and 3.12 when
$L_{1}(\Sigma,Y)$ is a $\mathcal{A}(L_{1}(\Sigma,Y))-C$-subspace of
$L_{1}(\Sigma,X)$, for subspaces $Y$ of $X$.

##  3\. Weighted restricted Chebyshev centers in spaces of vector-valued
functions

First, we note that the following may be established by the arguments given in
[1, Theorem 2].

######  Theorem 3.1.

Let $X$ be a Banach space which is uniformly convex and let $T$ be a
topological space. The Banach space $C_{b}(T,X)$ admits weighted Chebyshev
centers for all closed bounded subsets of $C_{b}(T,X)$ and bounded weights
$\rho$.

The following result demonstrates that in the spaces of Bochner integrable
functions $L_{p}(\Sigma,X)$, where $(\Omega,\Sigma,\mu)$ is a finite measure
space, the property $\mathscr{F}$-rcp is stable for a particular class of
functions $\mathscr{F}$. For $n\in\mathbb{N}$ and $1\leq p<\infty$,
$\mathscr{F}_{p}$ denotes a countable family of functions viz.
$(f_{p}^{n})_{n=1}^{\infty}$, where $f_{p}^{n}:\mathbb{R}^{n}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$ is defined by
$f_{p}^{n}(\alpha_{1},\cdots,\alpha_{n})=\left(\sum_{i=1}^{n}|\alpha_{i}|^{p}\right)^{\frac{1}{p}}$.

######  Theorem 3.2.

Let $Y$ be a separable subspace of $X$. Thus,

  * $(a)$

$(X,Y,\mathcal{F}(X))$ has _wrcp_ if and only if
$(L_{\infty}(\Sigma,X),L_{\infty}(\Sigma,Y),\mathcal{F}(L_{\infty}(\Sigma,X)))$
has _wrcp_.

  * $(b)$

for $1\leq p<\infty$, $(X,Y,\mathcal{F}(X))$ has _$\mathscr{F}_{p}$ -rcp_ if
and only if $(L_{p}(\Sigma,X),L_{p}(\Sigma,Y),\mathcal{F}(L_{p}(\Sigma,X)))$
has _$\mathscr{F}_{p}$ -rcp_.

###### Proof.

$(a)$. Let $F\in\mathcal{F}(L_{\infty}(\Sigma,X))$. We prove the result when
$|F|=2$ because new ideas are not involved for higher values of
$\textrm{card}(|F|)$. Let $F=\\{f_{1},f_{2}\\}$ and $\rho=(\rho_{1},\rho_{2})$
be the corresponding weights. Now, there exists a measurable subset
$E\subseteq T$ such that $\mu(E)=0$ and $f_{1},f_{2}$ are bounded on
$T\setminus E$. Define for all $t\in T\setminus E$,
$F_{t}=\\{f_{1}(t),f_{2}(t)\\}$. Let $G=\\{(t,y)\in(T\setminus E)\times
Y:r_{\rho}(y,F_{t})=\textrm{rad}_{Y}^{\rho}(F_{t})\\}$. Since
$(X,Y,\mathcal{F}(X))$ has wrcp, the projection of $G$ on $T\setminus E$ is
$T\setminus E$.

Suppose $(y_{n})$ is dense in $Y$. Then,
$G=\cap_{n\in\mathbb{N}}\\{(t,y)\in(T\setminus E)\times
Y:r_{\rho}(y,F_{t})\leq r_{\rho}(y_{n},F_{t})\\}$. Since all the involved
functions are measurable, $G$ is a measurable set. Thus, as a consequence of
the von Neumann selection theorem [15, Theorem 7.2], we obtain a measurable
function $g_{0}:(T\setminus E)\rightarrow Y$ such that $(t,g_{0}(t))\in G$ for
a.e $[\mu]$. Then, $g_{0}(t)\in\textrm{Cent}_{Y}^{\rho}(F_{t})$ for a.e. $t\in
T\setminus E$. Clearly, $g_{0}\in L_{\infty}(\Sigma,Y)$. For $g\in
L_{\infty}(\mu,X)$, we obtain

$\displaystyle r_{\rho}(g,F)$ $\displaystyle=$
$\displaystyle\underset{i=1,2}{\max}\leavevmode\nobreak\ \underset{t\in
T}{\textrm{ess\leavevmode\nobreak\ sup}}\rho_{i}\|g(t)-f_{i}(t)\|$
$\displaystyle=$ $\displaystyle\underset{t\in
T}{\textrm{ess\leavevmode\nobreak\ sup}}\leavevmode\nobreak\
\underset{i=1,2}{\max}\rho_{i}\|g(t)-f_{i}(t)\|$ $\displaystyle\geq$
$\displaystyle\underset{t\in T}{\textrm{ess\leavevmode\nobreak\
sup}}\leavevmode\nobreak\ \underset{i=1,2}{\max}\rho_{i}\|g_{0}(t)-f_{i}(t)\|$
$\displaystyle=$ $\displaystyle\underset{i=1,2}{\max}\leavevmode\nobreak\
\underset{t\in T}{\textrm{ess\leavevmode\nobreak\
sup}}\rho_{i}\|g_{0}(t)-f_{i}(t)\|$ $\displaystyle=$ $\displaystyle
r_{\rho}(g_{0},F).$

Hence, $g_{0}\in\textrm{Cent}^{\rho}_{L_{\infty}(\Sigma,Y)}(F)$.

$(b)$. Let $F\in\mathcal{F}(L_{p}(\Sigma,X))$. We prove the result when
$|F|=2$, as no new ideas are involved for higher values of
$\textrm{card}(|F|)$. Let $F=\\{f_{1},f_{2}\\}$ and
$f_{p}^{2}:\mathbb{R}_{\geq 0}^{2}\rightarrow\mathbb{R}_{\geq 0}$ represent a
function in $\mathscr{F}_{p}$. Accordingly, there exists a measurable subset
$E\subseteq T$ such that $\mu(E)=0$ and $f_{1},f_{2}$ are finite valued on
$T\setminus E$. Define for all $t\in T\setminus E$,
$F_{t}=\\{f_{1}(t),f_{2}(t)\\}$. Let $G=\\{(t,y)\in(T\setminus E)\times
Y:r_{f_{p}^{2}}(y,F_{t})=\textrm{rad}_{Y}^{f_{p}^{2}}(F_{t})\\}$. Since
$(X,Y,\mathcal{F}(X))$ has $\mathscr{F}_{p}$-rcp, the projection of $G$ on
$T\setminus E$ is $T\setminus E$.

Thus, in a similar manner as the proof of $(a)$, we obtain a measurable
function $g_{0}:(T\setminus E)\rightarrow Y$ such that
$(t,g_{0}(t))\in\textrm{Cent}_{Y}^{f_{p}^{2}}(F(t))$ a.e. $[\mu]$. Clearly,
$g_{0}\in L_{p}(\Sigma,Y)$. It is easy to see that
$g_{0}\in\textrm{Cent}_{L_{p}(\Sigma,Y)}^{f_{p}^{2}}(F)$. ∎

The following is a variation of [7, Proposition IV.1.2].

######  Theorem 3.3.

Let $X$ be an $L$-embedded space and $Y$ be a subspace of $X$, which is also
$L$-embedded and $F\in\mathcal{B}(X)$. Then,

  * $(a)$

for all continuous, monotone, and coercive functions
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$,
$\emph{Cent}_{Y}^{f}(F)\neq\emptyset$.

  * $(b)$

for all continuous, strictly monotone, and coercive functions
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$, $\emph{{Cent}}_{Y}^{f}(F)$ is
weakly compact.

###### Proof.

$(a)$. Let $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous,
monotone, and coercive function. Let $P:X^{**}\rightarrow X$ be the
$L$-projection. Thus, from [7, Theorem IV.1.2], we have
$P(\overline{Y}^{w^{*}})\subseteq Y$. There exists
$y^{**}\in\overline{Y}^{w^{*}}$ such that
$r_{f}(y^{**},F)=\textrm{rad}_{\overline{Y}^{w^{*}}}^{f}(F)$. Based on the
monotonicity of $f$, we obtain,

(1)
$\textrm{rad}_{Y}^{f}(F)\geq\textrm{rad}_{\overline{Y}^{w^{*}}}^{f}(F)=r_{f}(y^{**},F)\geq
r_{f}(Py^{**},F)\geq\textrm{rad}_{Y}^{f}(F).$

Since $Py^{**}\in Y$, our first assertion follows.

$(b)$. Let $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous,
strictly monotone, and coercive function. From the proof of $(a)$; equation 1,
we obtain a $y^{**}\in\overline{Y}^{w^{*}}$ such that,
$r_{f}(Py^{**},F)=r_{f}(y^{**},F).$ Now for all $x\in F$,

$\|y^{**}-x\|=\|Py^{**}-x+(Id-P)y^{**}\|=\|Py^{**}-x\|+\|y^{**}-Py^{**}\|.$

This leads to
$r_{f}(Py^{**},F)=f\left((\|Py^{**}-x\|+\|y^{**}-Py^{**}\|)_{x\in F}\right)$.

Since $f$ is strictly monotone, we have $y^{**}=Py^{**}$ and consequently,
$\textrm{Cent}_{Y}^{f}(F)=\textrm{Cent}_{\overline{Y}^{w^{*}}}^{f}(F)$. Hence,
the weak∗-closure of $\textrm{Cent}_{Y}^{f}(F)$ is contained in $X$, so it is
weakly compact. ∎

As a corollary of the Theorem 3.3, we have,

######  Corollary 3.4.

  * $(a)$

Let $X$ be an $L$-embedded space and $Y$ be a subspace of $X$, which is also
$L$-embedded. Suppose that $Y$ has $n.X.I.P.$ in $X$. Then $Y$ is a central
subspace of $X$.

  * $(b)$

Let $L_{1}(\Sigma,X)$ be an $L$-embedded space. Let
$F\in\mathcal{B}(L_{1}(\Sigma,X))$ and
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous, monotone, and
coercive function. Thus, for all sub $\sigma$-algebra
$\Sigma^{\prime}\subseteq\Sigma$,
$\emph{Cent}_{L_{1}(\Sigma^{\prime},X)}^{f}(F)\neq\emptyset$.

###### Proof.

$(a).$ From our assumption, it follows that $X(Y)$ admits weighted Chebyshev
center for finite subsets of $Y$. The result now follows from Corollary 2.9,
and Theorem 3.3.

$(b).$ Since $L_{1}(\Sigma^{\prime},X)$ is one complemented in
$L_{1}(\Sigma,X)$, it follows from [7, Proposition IV.1.5], that
$L_{1}(\Sigma^{\prime},X)$ is also $L$-embedded. Thus, our conclusion follows
from Theorem 3.3. ∎

It is relevant to mention that in some cases, the conclusion of Theorem 3.3(b)
is also valid if we drop the condition of strict monotonicity. For instance,
if $\rho:F\rightarrow\mathbb{R}_{>0}$ represents positive weights associated
with the elements of $F$, then $\textrm{Cent}_{Y}^{\rho}(F)$ also weakly
compact in $Y$. Similar arguments stated in the proof of Theorem 3.3(b) also
leads to the following.

######  Lemma 3.5.

Let $X$ be an $L$-embedded space and $Y$ be a subspace of $X$, which is also
$L$-embedded. Let $F\in\mathcal{B}(X)$ and
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous, strictly
monotone, coercive function. Suppose that $(y_{n})$ be a sequence in $Y$ such
that $\lim_{n}r_{f}(y_{n},F)=\emph{rad}_{Y}^{f}(F)$. Then, $(y_{n})$ is
relatively weakly compact.

As a consequence of Lemma 3.5, we have the following.

######  Theorem 3.6.

Let $X$ be a Banach space such that $L_{1}(\Sigma,X)$ is $L$-embedded. Let
$\\{\Sigma_{n}\\}_{i=1}^{\infty}$ be an increasing sequence of sub
$\sigma$-algebras. Let $\Sigma_{\infty}$ be the $\sigma$-algebra generated by
$\cup_{n\in\mathbb{N}}\Sigma_{n}$. Let $F\in\mathcal{B}(L_{1}(\Sigma,X))$ and
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous, strictly
monotone, and coercive function. Suppose that
$\emph{rad}_{L_{1}(\Sigma_{n},X)}^{f}(F)=r_{f}(f_{n},F)$ for some $f_{n}\in
L_{1}(\Sigma_{n},X)$. Then, $(f_{n})$ is relatively weakly compact, and any
weak limit point of this sequence belongs to
$\emph{Cent}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)$.

###### Proof.

We follow techniques similar to those used in [16, Theorem 12]. Let $E_{n}$ be
the conditional expectation projection of $L_{1}(\Sigma_{\infty},X)$ onto
$L_{1}(\Sigma_{n},X)$ for all $n\in\mathbb{N}$. It is known that
$E_{n}(g)\rightarrow g$ for all $g\in L_{1}(\Sigma_{\infty},X)$. We have for
any $n\in\mathbb{N}$,
$\textrm{rad}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)\leq\textrm{rad}_{L_{1}(\Sigma_{n},X)}^{f}(F)\leq
r_{f}(f_{n},F)$ and hence
$\textrm{rad}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)\leq\liminf_{n}r_{f}(f_{n},F)$.

Now for any $g\in L_{1}(\Sigma_{\infty},X)$, we have,

$\displaystyle\limsup_{n}\leavevmode\nobreak\
\textrm{rad}_{L_{1}(\Sigma_{n},X)}^{f}(F)\leq\lim_{n}r_{f}(E_{n}(g),F)\leq
r_{f}(g,F).$

Hence,
$\limsup_{n}r_{f}(f_{n},F)=\limsup_{n}\textrm{rad}_{L_{1}(\Sigma_{n},X)}^{f}(F)\leq\textrm{rad}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)$.

Thus,
$\textrm{rad}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)=\lim_{n}r_{f}(f_{n},F)=\lim_{n}\textrm{rad}_{L_{1}(\Sigma_{n},X)}^{f}(F)$.
Hence, by Lemma 3.5, $(f_{n})$ is relatively weakly compact. Using the lower
semi-continuity of the norm w.r.t. weak topology, we have the weak limit point
of $(f_{n})$ belonging to $\textrm{Cent}_{L_{1}(\Sigma_{\infty},X)}^{f}(F)$. ∎

The following Lemma is easy to prove.

######  Lemma 3.7.

Let $F\in\mathcal{B}(X)$ and $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be
continuous, monotone, and coercive function. Consequently, the function
defined by $r_{f}(x,F)=f((\|x-a\|)_{a\in F})$ is continuous and coercive.

######  Remark 3.8.

It is easy to observe that if $X$ is a dual space, then the above function
$r_{f}(.,F)$ is $w^{*}$-lower semicontinuous.

######  Proposition 3.9.

Suppose that $X$ is a dual space, $Y$ is a subspace of $X$,
$F\in\mathcal{B}(X)$ and $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a
continuous, monotone, and coercive function. Then,
$\emph{Cent}_{Y}^{f}(F)\neq\emptyset$.

###### Proof.

Follows from $w^{*}$ lower semi-continuity of $r_{f}(.,F)$. ∎

Our next observation follows from [18, Theorem 5.1] and the existence of the
conditional expectation operator.

######  Theorem 3.10.

Let $X$ denote a Banach space such that $X^{*}$ has $RNP$. Let $S$ be a
compact Hausdorff space and $(S,\Sigma,\mu)$ be a finite measure space. Let
$F\in\mathcal{B}(L_{1}(\Sigma,X^{*}))$ and
$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$ be a continuous, monotone, and
coercive function. Thus, for all sub $\sigma$-algebra
$\Sigma^{\prime}\subseteq\Sigma$,
$\emph{Cent}_{L_{1}(\Sigma^{\prime},X^{*})}(F)\neq\emptyset$.

Let us recall that $\mathcal{WC}(X)$ denotes the set of all weakly compact
subsets of $X$. Additionally, let us also recall the following theorem of
Dunford (see [6, pg. 101]).

######  Theorem 3.11.

Let $X$ be a reflexive Banach space. Then, $K\subseteq L_{1}(\mu,X)$ is
relatively weakly compact if and only if K is bounded and uniformly
integrable.

It is apparent from the proof of the above Theorem that for any Banach space
$X$, a relatively weakly compact subset of $L_{1}(\mu,X)$ is bounded and
uniformly integrable.

######  Theorem 3.12.

Let $Y$ be a reflexive subspace of $X$ and $(\Omega,\Sigma,\mu)$ be a finite
measure space.

  * $(a)$

Then for $F\in\mathcal{WC}(L_{1}(\Sigma,X))$, for all sub-$\sigma$-algebra
$\Sigma^{\prime}\subseteq\Sigma$ and for all continuous, monotone and coercive
function$f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$,
$\emph{Cent}_{L_{1}(\Sigma^{\prime},Y)}^{f}(F)\neq\emptyset$.

  * $(b)$

Then for all $F\in\mathcal{B}(L_{\infty}(\Sigma,X))$, and continuous, monotone
and coercive function $f:\Gamma(|F|)\rightarrow\mathbb{R}_{\geq 0}$,
$\emph{Cent}_{L_{\infty}(\Sigma,Y)}^{f}(F)\neq\emptyset$.

###### Proof.

$(a).$ Let $(g_{n})\subseteq L_{1}(\Sigma^{\prime},Y)$ be a such that
$r_{f}(g_{n},F)\rightarrow\textrm{rad}^{f}_{L_{1}(\Sigma^{\prime},Y)}(F)$.
Clearly, $(g_{n})$ is bounded. Hence by a vector-valued version of Kadec-
Pełczyns̀ki-Rosenthal theorem ([4, Lemma 2.1.3]), there exists a subsequence
$(g_{n_{k}})$ of $(g_{n})$ and a sequence of pairwise disjoint sets
$(A_{k})\subseteq\Sigma^{\prime}$ such that $(g_{n_{k}}\chi_{A^{c}_{k}})$ is
uniformly integrable in $L_{1}(\Sigma^{\prime},Y)$. Since,
$\sum_{k=1}^{\infty}\mu(A_{k})=\mu(\cup_{k=1}^{\infty}A_{k})\leq\mu(\Omega)<\infty$,
we have $\mu(A_{k})\rightarrow 0$.

Let us choose $h\in F$ then,

$\|g_{n_{k}}\chi_{A_{k}^{c}}-h\|_{1}=\int_{A_{k}}\|h\|d\mu+\int_{A_{k}^{c}}\|g_{n_{k}}-h\|d\mu\leq\int_{A_{k}}\|h\|d\mu+\|g_{n_{k}}-h\|_{1}.$

As $F$ is uniformly integrable, we get $\lim_{k}\int_{A_{k}}\|h\|d\mu=0$
uniformly for all $h\in F$. Now, it is easy to see that
$r_{f}(g_{n_{k}}\chi_{A_{k}^{c}},F)\rightarrow\textrm{rad}^{f}_{L_{1}(\mathcal{B},Y)}(F)$.
Also since $(g_{n_{k}}\chi_{A_{k}^{c}})$ is a bounded sequence in
$L_{1}(\Sigma^{\prime},Y)$, by Theorem 3.11, $(g_{n_{k}}\chi_{A_{k}^{c}})$ is
relatively weakly compact in $L_{1}(\Sigma^{\prime},Y)$. Let us denote by
$(g_{n_{k(j)}})$, a weakly convergent subsequence of
$(g_{n_{k}}\chi_{A_{k}^{c}})$ converging weakly to $g\in
L_{1}(\Sigma^{\prime},Y)$. Then $r_{f}(g,F)\leq\lim\inf
r_{f}(g_{n_{k(j)}},F)=\textrm{rad}_{L_{1}(\Sigma^{\prime},Y)}^{f}(F)$. Hence
$r_{f}(g,F)=\textrm{rad}^{f}_{L_{1}(\Sigma^{\prime},Y)}(F)$.

$(b).$ We consider $L_{\infty}(\Sigma,X)$ to be canonically embedded in
$L_{1}(\Sigma,X^{*})^{*}$. It follows from from Proposition $3.1$ of [14] that
$L_{\infty}(\Sigma,Y)$ is weak∗-closed in $L_{1}(\Sigma,X^{*})^{*}$. Hence by
the weak∗ semicontinuity of the norm, we have an $g\in L_{\infty}(\Sigma,Y)$
such that $r_{f}(g,F)=\textrm{rad}_{L_{\infty}(\Sigma,Y)}^{f}(F)$. ∎

######  Remark 3.13.

Using similar arguments, it can be shown that for a reflexive subspace $Y$ of
$X$ and for a sub-$\sigma$-algebra $\Sigma^{\prime}\subseteq\Sigma$,
$(L_{1}(\Sigma,X),B_{L_{1}(\Sigma^{\prime},Y)},\mathcal{WC}(L_{1}(\Sigma,X)))$
has _rcp_.

Let $\alpha$ be an ordinal where $\alpha=\omega_{0}$ or $\omega_{1}$, the
first infinite ordinal and the first uncountable ordinal respectively. Let
$\mathscr{F}_{\alpha}$ be a family of continuous monotone functions
$f:\Gamma(I)\rightarrow\mathbb{R}_{\geq 0}$, where $I\leq\alpha$.

######  Proposition 3.14.

Let $P$ be a projection on $X$ of norm one and $Y=\ker(P)$. Let $Z\subseteq
P(X)$ be a subspace. Let $\mathcal{F}_{1}^{\alpha}(\mathcal{F}_{2}^{\alpha})$
be the family of subsets $F$ of $\mathcal{B}(X)(\mathcal{B}(P(X)))$ with
_card_ $(F)\leq\alpha$. If $(X,Y+Z,\mathcal{B}(X))$ has
_$\mathscr{F}_{\alpha}$ -rcp_, then $(P(X),Z,\mathcal{B}(P(X)))$ has
_$\mathscr{F}_{\alpha}$ -rcp_.

###### Proof.

Let $F\in\mathcal{F}_{2}^{\alpha}$ and $f\in\mathscr{F}_{\alpha}$. Based on
our assumption, there exists $y_{0}+z_{0}\in Y+Z$ such that
$r_{f}(y_{0}+z_{0},F)=\textrm{rad}_{Y+Z}^{f}(F)\leq\textrm{rad}_{Z}^{f}(F)$.
Then, $r_{f}(z_{0},F)=r_{f}(P(z_{0}+y_{0}),F)\leq
r_{f}(z_{0}+y_{0},F)\leq\textrm{rad}_{Z}^{f}(F)$. ∎

######  Remark 3.15.

Similarly, with the same conditions as that in Proposition 3.14, it can be
shown that if $(X,B_{Y+Z},\mathcal{B}(X))$ has rcp, then
$(P(X),B_{Z},\mathcal{B}(P(X)))$ has rcp.

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2(\frac{2}{c_{\mu}\mu^{-}\pi_{d}})^{\frac{1}{d}}(\frac{k_{Q}}{n_{Q}})^{\frac{1}{d}}.$

This completes the proof by setting $c_{D}$ large enough. ∎

######  Lemma 4.

For any empirical classifier $\hat{f}$ and $\alpha,\beta\in\mathbb{R}^{d}$
such that $\|\alpha\|$ and $\|\beta\|$ are bounded between $c$ and $C$ for
some constant $C>c>0$, and $\angle(\alpha,\beta)\in[0,\pi/2]$. Then we have

$\mathcal{E}_{Q_{\alpha}}(\hat{f})+\mathcal{E}_{Q_{\beta}}(\hat{f})\geq\frac{c\sigma(C)(1-\sigma(C))}{20\pi}\angle(\alpha,\beta)^{2}.$

###### Proof.

Without loss of generality and due to the symmetry property of of
$N(0,I_{d})$, we could rotate $\alpha$ and $\beta$ at the same time so that

$\alpha=(\|\alpha\|,0,0,\cdots,0),\quad\beta=(\|\beta\|\cos\angle(\alpha,\beta),\|\beta\|\sin\angle(\alpha,\beta),0,0,\cdots,0).$

Therefore, we assume that only the first coordinate of $\alpha$ and the first
and second coordinates of $\beta$ can be non-zero.

For any $\lambda\in[0,1]$, we define
$w_{\lambda}:=\lambda\alpha+(1-\lambda)\beta$. Define
$\lambda_{0},\lambda_{1}\in[0,1]$ as the value satisfying that

$\angle(\alpha,w_{\lambda_{0}})=\angle(w_{\lambda_{1}},\beta)=\angle(\alpha,\beta)/4.$

For simplicity, we abbreviate $w_{\lambda_{0}}$ as $w_{0}$ and
$w_{\lambda_{1}}$ as $w_{1}$. Define the area

$D:=\\{x\in\mathbb{R}^{d}:(w_{0}^{T}x)(w_{1}^{T}x)<0,\frac{1}{4}\leq
x_{1}^{2}+x_{2}^{2}\leq 1\\}.$

It is easy to see that
$D\subset\\{x\in\mathbb{R}^{d}:(\alpha^{T}x)(\beta^{T}x)<0,\frac{1}{2}\leq
x_{1}^{2}+x_{2}^{2}\leq 1\\}$. Moreover, we see that if $x\in D$, we have

$|\alpha^{T}x|=\|\alpha\|x_{1}\geq\|\alpha\|\sin(\frac{\angle(\alpha,\beta)}{4})(x^{1}+x^{2})^{\frac{1}{2}},$
$|\beta^{T}x|\geq\|\beta\|\sin(\frac{\angle(\alpha,\beta)}{4})(x^{1}+x^{2})^{\frac{1}{2}}$

since the angle between $(x_{1},x_{2},0\cdots,0)$ and normal planes of
$\alpha$ and $\beta$ are both in
$[\frac{\angle(\alpha,\beta)}{4},\frac{3\angle(\alpha,\beta)}{4}]$. Therefore,
since $\sigma^{\prime}(t)=\sigma(t)(1-\sigma(t))$ decreases with growing
$|t|$, and $|\alpha^{T}x|\leq\|\alpha\|\leq C$ on $x\in D$, we have that if
$x\in D$.

$\displaystyle|\eta^{Q_{\alpha}}(x)-\frac{1}{2}|$
$\displaystyle\geq|\sigma(\sin(\frac{\angle(\alpha,\beta)}{4})\|\alpha\|(x_{1}^{2}+x_{2}^{2})^{\frac{1}{2}})-\frac{1}{2}|$
$\displaystyle\geq\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4})\|\alpha\|(x_{1}^{2}+x_{2}^{2})^{\frac{1}{2}}$
$\displaystyle\geq\frac{1}{2}c\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4}).$

Similarly,

$|\eta^{Q_{\beta}}(x)-\frac{1}{2}|\geq\frac{1}{2}c\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4}).$

Hence, we have

$\displaystyle\mathcal{E}_{Q_{\alpha}}(\hat{f})+\mathcal{E}_{Q_{\beta}}(\hat{f})$
$\displaystyle=2\mathbb{E}_{Q_{\alpha}}[|\eta^{Q_{\alpha}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\alpha^{T}x)<0\\}]$
$\displaystyle+2\mathbb{E}_{Q_{\beta}}[|\eta^{Q_{\beta}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\beta^{T}x)<0\\}]$
$\displaystyle\geq
2\mathbb{E}_{Q_{\alpha}}[|\eta^{Q_{\alpha}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\alpha^{T}x)<0,x\in
D\\}]$
$\displaystyle+2\mathbb{E}_{Q_{\beta}}[|\eta^{Q_{\beta}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\beta^{T}x)<0,x\in
D\\}]$ $\displaystyle\geq
c\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4})\times$
$\displaystyle\mathbb{E}_{X\sim
N(0,I_{d})}[\left(\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\alpha^{T}x)<0\\}+\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\beta^{T}x)<0\\}\right)\mathbf{1}\\{x\in
D\\}].$

The change of the expectation operator in the last inequality is due to the
fact that
$|\eta^{Q_{\alpha}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\alpha^{T}x)<0\\}$
and
$|\eta^{Q_{\beta}}(X)-\frac{1}{2}|\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\beta^{T}x)<0\\}$
do not depend on the distribution of the response value.

If $x\in D$, then $(\alpha^{T}x)(\beta^{T}x)<0$, which implies

$\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\alpha^{T}x)<0\\}+\mathbf{1}\\{(\hat{f}-\frac{1}{2})(\beta^{T}x)<0\\}=1.$

Therefore, we could further bound
$\mathcal{E}_{Q_{\alpha}}(\hat{f})+\mathcal{E}_{Q_{\beta}}(\hat{f})$ by

$\displaystyle\mathcal{E}_{Q_{\alpha}}(\hat{f})+\mathcal{E}_{Q_{\beta}}(\hat{f})$
$\displaystyle\geq
c\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4})Q_{X}(D)$
$\displaystyle\geq
c\sigma(C)(1-\sigma(C))\sin(\frac{\angle(\alpha,\beta)}{4})\angle(\alpha,\beta)Q(X_{1}^{2}+X_{2}^{2}\in[\frac{1}{4},1])$
$\displaystyle\geq\frac{c\sigma(C)(1-\sigma(C))}{20\pi}\angle(\alpha,\beta)^{2}.$

where $\Psi(\cdot)$ is the cumulative distribution function of the univariate
standard normal distribution. Note that we utilized the numerical results that

$\sin(\frac{\angle(\alpha,\beta)}{4})\geq\frac{1}{2\pi}\angle(\alpha,\beta),$

and the cumulative distribution function of chi-squared distribution with $2$
degrees of freedom

$Q(X_{1}^{2}+X_{2}^{2}\in[\frac{1}{4},1])=e^{-\frac{1}{2}}-e^{-\frac{1}{8}}\geq\frac{1}{10}.$

∎