The almost fixed point property is not invariant under isometric renormings

Abstract

In the present note we prove the non set-stability of the AFPP under isometric renormings in the setting of Banach spaces containing a complemented isomorphic copy of $c_0$ or $\ell _1$

1 Introduction

Let X be a Banach space which has a property ${\mathcal {P}}$. One of the fundamental problems in metric fixed point theory is to determine if ${\mathcal {P}}$ is stable, that is, if the property ${\mathcal {P}}$ is shared by all the spaces ‘near enough’ to X. To make precise the problem of stability usually we consider the following definition of distance introduced by Stefan Banach in [1] and known as Banach-Mazur distance.

Definition 1

Let X and Y be Banach spaces. The Banach-Mazur distance between X and Y, denoted by d(X, Y) is defined as:

$$\begin{aligned} d(X,Y)={\mathrm { inf}}\,\{ \Vert T\Vert \cdot \Vert T^{-1} \Vert : {T} \text { is an isomorphism from } X \text { to} Y\}. \end{aligned}$$

When X and Y are not isomorphic, we say that $d(X,Y)=\infty$.

We say that ${\mathcal {P}}$ is stable if for every Banach space X which satisfies ${\mathcal {P}}$, there exists $\gamma >0$ such that $d(X,Y)<\gamma$ implies that Y verifies ${\mathcal {P}}$. All those properties that are invariant under isomorphism are stable in this sense, for instance reflexivity, super reflexivity, Banach-Saks property, Krein-Milman property, Radon-Nykodým property, Shur property, property of being a weakly compactly generated space, stable weak$^*$ fixed point property ([2, 3]), uniform normal structure, uniform non-squareness among others.

Since by definition $d(X,Y)\ge 1$ we can restrict the mentioned notion of stability to the case in which $d(X,Y)=1$.

Definition 2

We say that a property ${\mathcal {P}}$ is invariant under Banach-Mazur distance 1 if for every pair of Banach spaces X, Y, with $d(X,Y)=1$, X has the property ${\mathcal {P}}$ if and only if Y has the property ${\mathcal {P}}$.

In [17] Ł. Piasecki proved that the weak$^*$ fixed point property ($w^*$-fpp), $w^*$-normal structure ($w^*$-NS), the weak$^*$ Kadec-Klee property ($w^*$-KK), the weak$^*$ Generalized Gossez-Lami Dozo property ($w^*$-GGLD), the finite dimensional norm preserving extension property (FNEP) and the compact norm preserving extension property (CNEP) are not invariant under Banach-Mazur distance 1, even in the framework of separable Lindenstrauss spaces. In the general setting of Banach spaces, he proved also that uniform convexity in every direction, local uniform rotundity, uniform smoothness, the Kadec-Klee property, the weak$^*$-Opial property, the Opial property, the weak fixed point property and weak normal structure are not invariant under Banach-Mazur distance 1.

Let C be a closed convex subset of a Banach space X and $T:C\rightarrow C$ a nonexpansive function, that is, for every $x,y\in C$, $\Vert Tx-Ty\Vert \le \Vert x-y\Vert$. Whenever $\mathrm {inf}\,\{ \Vert Tx-x\Vert :x\in C\}=0$, we say that C satisfies the almost fixed point property (AFPP). In relation to this concept, in [9] it was studied the following notion of stability:

Definition 3

We denote by ${\mathcal {C}}(X,\Vert \cdot \Vert )$ the collection of closed convex sets with the AFPP in $(X,\Vert \cdot \Vert )$. We will say that $(X,\Vert \cdot \Vert )$ has stability of the AFPP if for any norm $\Vert \cdot \Vert _1$ on X equivalent to $\Vert \cdot \Vert$, we have that ${\mathcal {C}}(X,\Vert \cdot \Vert )={\mathcal {C}}(X,\Vert \cdot \Vert _1)$. If $\Vert \cdot \Vert _1$ is a norm on X equivalent to $\Vert \cdot \Vert$, we say that the collections of sets with the AFPP in $(X,\Vert \cdot \Vert )$ and $(X,\Vert \cdot \Vert _1)$ differ when ${\mathcal {C}}(X,\Vert \cdot \Vert )\ne {\mathcal {C}}(X,\Vert \cdot \Vert _1)$.

Goebel and Kuczumow in [13] proposed the problem of characterizing those closed convex subsets of a Banach space verifying the AFPP. Regarding this, a result obtained by S. Reich in [18] establishes that if X is a reflexive space, a closed convex set $C\subset X$ has the AFPP if and only if C is linearly bounded (it contains no ray). Observe that the condition of linearly boundedness is invariant under isomorphisms and consequently if $(X,\Vert \cdot \Vert )$ is a reflexive space, X has stability of the AFPP.

Afterwards in [19] I. Shafrir introduced the concept of directionally boundedness and proved, without assumptions of reflexivity, that a closed convex subset C of a Banach space X has the AFPP if and only if C is directionally bounded. In the last years techniques involving the AFPP have been very useful in the study of various problems in fixed point theory [4, 7, 8, 10, 15].

Recently in [9] the authors proved the following characterization of reflexivity in terms of the AFPP:

Theorem 1

Let X be a Banach space. Then X is reflexive if and only if X has stability of the AFPP. Moreover, if $(X,\Vert \cdot \Vert )$ is not reflexive, for every $\delta >0$, there exists an equivalent norm $\vert \cdot \vert$ in X such that $d((X,\Vert \cdot \Vert ),(X,\vert \cdot \vert ))<1+\delta$ and the respective collections of sets with the AFPP differ.

Considering the problem posed by Piasecki in [17] of determining geometrical properties that are not invariant under Banach-Mazur distance 1, we propose the following strengthening of definition 1.1 in [9]:

Definition 4

We say that a Banach space $(X,\Vert \cdot \Vert )$ has set-stability of the AFPP under isometric renormings, if for every norm $\vert \cdot \vert$ on X such that $(X,\Vert \cdot \Vert )$ and $(X,\vert \cdot \vert )$ are isometric spaces, we have that ${\mathcal {C}}(X,\Vert \cdot \Vert )={\mathcal {C}}(X,\vert \cdot \vert ).$

Observe that every reflexive Banach space has set-stability of the AFPP under isometric renormings.

Bearing in mind Definition 4 and the last conclusion in Theorem 1, we can ask if any Banach space has set-stability of the AFPP under isometric renormings.

The purpose of the present note is to give a negative answer to this question by showing a wide class of spaces which have not set-stability of the AFPP under isometric renormings.

2 Preliminaries

In [19] I. Shafrir introduced the following concept:

Definition 5

A sequence $(x_n)$ in a Banach space X is called a directional sequence if:

• (i) $\Vert x_n\Vert \rightarrow \infty$

• (ii) There is $b\ge 0$ such that for all $n_1<n_2<\cdots <n_l$

$$\begin{aligned} \Vert x_{n_1}-x_{n_l}\Vert \ge \sum _{i=1}^{l-1} \Vert x_{n_i}-x_{n_{i+1}}\Vert -b. \end{aligned}$$

He called a closed convex set $C\subset X$ directionally bounded if it contains no directional sequences and proved the following criterium to determine when C verifies this property.

Theorem 2

A closed convex set C in a Banach space X is directionally bounded if for every sequence $\{x_n\}$ in C such that $\Vert x_n\Vert \rightarrow \infty$ and for every f in the unitary ball of $X^*$,

$$\begin{aligned} \limsup _n f(x_n/\Vert x_n\Vert ) <1. \end{aligned}$$

In the same work he showed that the condition of directionally boundedness is equivalent to the AFPP in Banach spaces.

Theorem 3

A closed convex set C in a Banach space X has the AFPP if and only if it is directionally bounded.

Using some of the tools developed by Shafrir in his study of the AFPP, in [9] the authors proved the following characterization for reflexive spaces in terms of the AFPP.

Theorem 4

Let $(X,\Vert \cdot \Vert )$ be a non-reflexive Banach space. Then for every $\epsilon >0$ there is a renorming $(X,\vert \cdot \vert )$ with $d((X,\Vert \cdot \Vert ),(X,\vert \cdot \vert ))<1+\epsilon$ such that the respective families of closed convex subsets with the AFPP differ.

The space of sequences of summable modulus and the space of sequences converging to 0 are denoted respectively by $\ell _1$ and $c_0$. If we write $\ell _1$ or $c_0$ it is understood that $\ell _1=(\ell _1,\Vert \cdot \Vert _1)$ and $c_0=(c_0,\Vert \cdot \Vert _\infty )$ where $\Vert x\Vert _1=\sum _{i=1}^\infty \vert x(i)\vert$ and $\Vert x\Vert _\infty =\underset{i\in \mathbb {N}}{\mathrm {sup}}\, \vert x(i)\vert$.

The following results give conditions under which a (isomorphic) copy of $c_0$ or $\ell _1$ is complemented in a Banach space X. We include them in order to exemplify some applications of our main result. The first result is an immediate consequence of Proposition 1.8 in [14].

Theorem 5

Let X be a Banach space X with an unconditional basis. Then every copy of $\ell _1$ in X contains a complemented copy of $\ell _1$ (complemented in X).

Theorem 6

([5, Theorem 2.2]) Let X be a real Banach space that does not contain a copy of $\ell _1$. If X contains a copy of $c_0$, then X contains a complemented isomorphic copy of $c_0$.

Theorem 7

([6, Theorem 10 p. 48]) Let X be a Banach space. Then $X^*$ contains an isomorphic copy of $c_0$ if and only if X contains a complemented isomorphic copy of $\ell _1$.

Theorem 8

([12, Corollary 2.2]) Let X be a separable Banach space and let Y be a subspace of X which is isomorphic to $c_0$. Then Y is complemented in X.

Throughout this work we consider only real Banach spaces. We denote by ${\mathcal {P}}(X)$ the collection of equivalent norms to a fixed norm of a Banach space X.

3 Non stability of the AFPP under Banach-Mazur distance 1

We start proving our main result for the particular cases of $\ell _1$ and $c_0$.

Lemma 1

Let $(X,\Vert \cdot \Vert )=\ell _1$ or $(X,\Vert \cdot \Vert )=c_0$. Then there is a norm $\Vert \cdot \Vert _1 \in {\mathcal {P}}(X)$ such that $(X,\Vert \cdot \Vert _1)$ has not set-stability of the AFPP under isometric renormings.

Proof

Let $(e_n)$ denote the canonical basis of X and let $W=[e_{2n-1}]$, where $[e_{2n-1}]$ denotes the closed linear span of $\{ e_{2n-1}:n\in \mathbb {N}\}$. By Theorem 4 there is an equivalent norm $\vert \Vert \cdot \Vert \vert \in {\mathcal {P}}(W)$ such that the collections of directionally bounded sets in $(W,\Vert \cdot \Vert )$ and $(W,\vert \Vert \cdot \Vert \vert )$ differ, so without loss of generality we may assume that there is a closed convex unbounded set $C\subset W$ which is directionally bounded with respect to the norm $\Vert \cdot \Vert$ but it is not with respect to the norm $\vert \Vert \cdot \Vert \vert$. Let $P:X\rightarrow W$ be such that if $x=\sum _{n=1}^\infty a_n e_n\in X$, then $Px=\sum _{n=1}^\infty a_{2n-1}e_{2n-1}$. Define $\Vert \cdot \Vert _1$ and $\Vert \cdot \Vert _2\in {\mathcal {P}}(X)$ such that

$$\begin{aligned} \Vert x\Vert _1= \Vert Px\Vert +\vert \Vert (I-P)x\Vert \vert \end{aligned}$$

and

$$\begin{aligned} \Vert x\Vert _2= \vert \Vert Px\Vert \vert +\Vert (I-P)x\Vert . \end{aligned}$$

Since the inclusions $i_1: (W,\Vert \cdot \Vert )\rightarrow (X,\Vert \cdot \Vert _1)$ and $i_2:(W,\vert \Vert \cdot \Vert \vert )\rightarrow (X,\Vert \cdot \Vert _2)$ are isometries on their images, C has the AFPP with respect to the norm $\Vert \cdot \Vert _1$, but it has not the AFPP for the norm $\Vert \cdot \Vert _2$. Observe that if $x=\sum _{n=1}^\infty a_ne_n\in X,$ the function $T: (X,\Vert \cdot \Vert _1)\rightarrow (X,\Vert \cdot \Vert _2)$ defined as

$$\begin{aligned} (Tx)(i)= {\left\{ \begin{array}{ll} a_{2j-1}, &{} \text { if} i=2j \text { for some } j\in \mathbb {N},\\ a_{2j}, &{} \text {if } i=2j-1 \text { for some } j\in \mathbb {N}, \end{array}\right. } \end{aligned}$$

is an onto linear isometry. $\square$

We can generalize the conclusion in Lemma 1 to the class of spaces containing a complemented subspace isomorphic to $c_0$ or $\ell _1$.

Theorem 9

Let $(X,\Vert \cdot \Vert )$ be a Banach space containing a complemented isomorphic copy of $c_0$ or $\ell _1$. Then there is $\Vert \cdot \Vert _1\in {\mathcal {P}}(X)$ such that $(X,\Vert \cdot \Vert _1)$ has not set-stability of the AFPP under isometric renormings.

Proof

Let V be a complemented subspace of X isomorphic to Y, where Y is $c_0$ or $\ell _1$ and denote by $\vert \cdot \vert _1$, $\vert \cdot \vert _2$ the two norms on Y, garanteed by Lemma 1, such that the families of sets with the AFPP in $(Y,\vert \cdot \vert _1)$ and $(Y,\vert \cdot \vert _2)$ differ. Let $T_1: V\rightarrow (Y,\vert \cdot \vert _1)$ and $T_2:V\rightarrow (Y,\vert \cdot \vert _2)$ be isomorphisms. If we consider on V the norms $\vert \Vert \cdot \Vert \vert _1$ and $\vert \Vert \cdot \Vert \vert _2\in {\mathcal {P}}(V)$ such that

$$\begin{aligned} \vert \Vert x\Vert \vert _1=\vert T_1 x\vert _1\quad \text {and}\quad \vert \Vert x\Vert \vert _2=\vert T_2 x\vert _2, \end{aligned}$$

then $R_1: (V, \vert \Vert \cdot \Vert \vert _1) \rightarrow (Y,\vert \cdot \vert _1)$ and $R_2:(V, \vert \Vert \cdot \Vert \vert _2)\rightarrow (Y,\vert \cdot \vert _2)$ such that $R_i x=T_i x$, $i=1,2,$ are linear onto isometries. Let $T:(Y,\vert \cdot \vert _1)\rightarrow (Y,\vert \cdot \vert _2)$ be the isometry considered in Lemma 1. Since T is an onto linear isometry, we have that $L: (V, \vert \Vert \cdot \Vert \vert _1)\rightarrow (V, \vert \Vert \cdot \Vert \vert _2)$ defined as $L=R_2^{-1}TR_1$ is an onto linear isometry. Let $P: X\rightarrow V$ be a linear bounded projection from X onto V and define $\Vert \cdot \Vert _1$, $\Vert \cdot \Vert _2\in {\mathcal {P}}(X)$ such that

$$\begin{aligned} \Vert x\Vert _1= \vert \Vert P x\Vert \vert _1+\Vert (I-P)x\Vert \end{aligned}$$

and

$$\begin{aligned} \Vert x\Vert _2= \vert \Vert Px\Vert \vert _2+\Vert (I-P)x\Vert . \end{aligned}$$

If $S: (X,\Vert \cdot \Vert _1)\rightarrow (X,\Vert \cdot \Vert _2)$ is defined as $Sx=(I-P)x+LPx$ it is easy to check that S is an isomorphism and furthermore

$$\begin{aligned} \Vert Sx\Vert _2&= \vert \Vert PLP x\Vert \vert +\Vert (I-P)x\Vert =\vert \Vert LPx\Vert \vert _2+\Vert (I-P)x\Vert \\&= \vert \Vert Px\Vert \vert _1+\Vert (I-P)x\Vert =\Vert x\Vert _1, \end{aligned}$$

since L is an isometry. Cleary ${\mathcal {C}}( X,\Vert \cdot \Vert _1)\ne {\mathcal {C}}(X,\Vert \cdot \Vert _2)$. $\square$

Below we show a wide variety of cases in which we can apply Theorem 9.

Lemma 2

Let X be a non reflexive Banach space that satisfies any of the following conditions:

• (i) X has an unconditional Schauder Basis.

• (ii) X contains an isomorphic copy of $c_0$ and X does not contain an isomorphic copy of $\ell _1$.

• (iii) X is separable and contains an isomorphic copy of $c_0$.

• (iv) $X^*$ contains an isomorphic copy of $c_0$.

• (v) X has an uncountable unconditional Schauder basis [11] and contains an isomorphic copy of $\ell _1$.

Then there is a norm $\Vert \cdot \Vert _1\in {\mathcal {P}}(X)$ such that $(X,\Vert \cdot \Vert _1)$ has not set-stability of the AFPP under isometric renormings.

Proof

(i) By James theorem ([16, Corollary 4.4.23]) X contains a subspace Y isomorphic to $c_0$ or $\ell _1$. If Y is isomorphic to $c_0$, Theorem 8 implies that Y is complemented. If Y is isomorphic to $\ell _1$, by Theorem 5 we can assume that Y is complemented. So in either case, the hypotheses of Theorem 9 are satisfied. Similarly we prove (ii)–(v) using Theorem 6, Theorem 8, Theorem 7 and Theorem 1a in [11] respectively. $\square$

Lemma 2 offers a large class of examples of Banach spaces in which we can conclude the non set-stability of the AFPP under isometric renormings. However, it is still possible that given a Banach space X, we can find two non-isometric norms $\Vert \cdot \Vert _1, \Vert \cdot \Vert _2$ in ${\mathcal {P}}(X)$, such that $(X,\Vert \cdot \Vert _1)$ and $(X,\Vert \cdot \Vert _2)$ share the collection of sets with the AFPP and $d((X,\Vert \cdot \Vert _1),(X,\Vert \cdot \Vert _2))=1$. In the following, we illustrate this situation in the particular case of $X=c_0$.

Lemma 3

Consider the spaces $X=(c_0,\Vert \cdot \Vert )$ and $Y=(c_0,\vert \Vert \cdot \Vert \vert )$ where

$$\begin{aligned} \Vert x\Vert =\Vert x\Vert _\infty +\sum _{j=1}^\infty \frac{\vert x(j)\vert }{2^j} \end{aligned}$$

and

$$\begin{aligned} \vert \Vert x\Vert \vert =\Vert x\Vert _\infty +\sum _{j=2}^\infty \frac{\vert x(j)\vert }{2^{j-1}} \end{aligned}$$

then $d(X,Y)=1$.

Proof

Let $T_n:X\rightarrow Y$ be such that

$$\begin{aligned} T_n(x)=(x(n),x(1),\ldots , x(n-1), x(n+1),\ldots ). \end{aligned}$$

It is easy to see that $T_n$ is an isomorphism such that $\Vert T_n\Vert \rightarrow 1$ and $\Vert T^{-1}_n\Vert \rightarrow 1$. $\square$

We want to compare the collections of directionally bounded sets of the spaces X and Y defined in Lemma 3. In order to do this, by Theorem 2, it is very useful to know $X^*$ and $Y^*$. The space $X^*$ was described in [4], so we proceed to determine $Y^*$. Straightforward calculations allow us to prove the following result:

Proposition 1

Let $N\in \mathbb {N}$, $Z_N=\{ x\in Y: x(i)=0,\; \forall i>N\}$ and $Y_N=\left( Z_N,\vert \Vert \cdot \Vert \vert _{\vert _{Z_N}}\right)$. Then the set of extreme points with non-negative coordinates in the unitary ball of $Y_N$ $\left( \hbox { denoted as}\ \xi ^+(Y_N)\right)$ satisfies:

$$\begin{aligned} \xi ^+(Y_N)\subset \left\{ \left( \dfrac{1}{\sum _{i\in F}\frac{1}{2^{i-1}}}\right) \sum _{i\in F} e_i: F\subset \{1,\ldots , N\},\; 1\in F \right\} . \end{aligned}$$

Where $(e_n)$ is the canonical basis of $c_0$.

From the last proposition it follows that:

Proposition 2

$(c_0,\vert \Vert \cdot \Vert \vert )^*=(\ell _1,\Vert \cdot \Vert _*)$ where for $f=(c_i)\in \ell _1$

$$\begin{aligned} \Vert f\Vert _*= \underset{\begin{array}{c} F\subset \mathbb {N}\\ 1\in F,\; \# F<\infty \end{array}}{\mathrm {sup}}\left( \frac{1}{\sum _{i\in F}\frac{1}{2^{i-1}}}\right) \sum _{i\in F}\vert c_i\vert . \end{aligned}$$

The next proposition gives a criterium to distinguish between directional and non directional sequences in the space $(c_0,\vert \Vert \cdot \Vert \vert )$. Bearing in mind Proposition 2, the proof of this result is analogous to the proof of proposition 9 in [4].

Proposition 3

Let C be a closed convex unbounded set in $c_0$. Suppose that C is directionally bounded in $(c_0,\vert \Vert \cdot \Vert \vert )$. Let $(x_n)\subset C$ be a sequence such that $\lim _{n\rightarrow \infty }\Vert x_n\Vert _\infty =\infty$, with $x_n=(x_n(k))_{k=1}^\infty$. Then for every $n_0,k_0\in \mathbb {N}$ there exist $n>n_0$ and $k>k_0$ such that $\Vert x_n\Vert _\infty =\vert x_n(k)\vert$.

The following result describes a sufficient condition in $c_0$ for an unbounded sequence not to be directional.

Proposition 4

(Proposition 10 in [4]) Let $(x_n)\subset c_0$ be a sequence such that $\lim _{n\rightarrow \infty }\Vert x_n\Vert _\infty =\infty$. If for every $n_0,k_0\in \mathbb {N}$ there exists $n>n_0$ and $k>k_0$ such that $\Vert x_n\Vert _\infty =\vert x_n(k)\vert$, then $(x_n)$ is not a directional sequence in $(c_0,\Vert \cdot \Vert _\infty )$.

Using Propositions 3 and 4 we can establish a relationship between the collections of directionally bounded sets in $(c_0,\vert \Vert \cdot \Vert \vert )$ and $c_0$.

Theorem 10

Let C be a convex, closed and unbounded subset of $c_0$. Then C is directionally bounded in $(c_0,\Vert \cdot \Vert _\infty )$ if and only if it is directionally bounded in $(c_0,\vert \Vert \cdot \Vert \vert )$.

Proof

Let $C\subset (c_0,\vert \Vert \cdot \Vert \vert )$ be a closed convex unbounded and directionally bounded set and $(x_n)\subset C$ a sequence such that $\vert \Vert x_n\Vert \vert \rightarrow \infty$. By Proposition 3 for every $n_0, k_0\in \mathbb {N}$ there are $n>n_0$ and $k>k_0$ such that $\Vert x_n\Vert _\infty = \vert x_n(k)\vert$ and Proposition 4 implies that $(x_n)$ is not a directional sequence in $(c_0,\Vert \cdot \Vert _\infty ).$

Conversely, suppose that $C\subset (c_0,\Vert \cdot \Vert _\infty )$ is a closed convex and unbounded set. If $(x_n)$ is a directional sequence in $(c_0,\vert \Vert \cdot \Vert \vert )$. Take $b\ge 0$ as given by the definition of directional sequence. Let $n_1<n_2<\cdots <n_s.$ Then

$$\begin{aligned} b\ge & {} \sum _{i=1}^{s-1}\vert \Vert x_{n_i}-x_{n_{i+1}}\Vert \vert -\vert \Vert x_{n_s}-x_{n_1}\Vert \vert \\= & {} \sum _{i=1}^{s-1}\Vert x_{n_i}-x_{n_{i+1}}\Vert _\infty -\Vert x_{n_s}-x_{n_1}\Vert _\infty \\&+\sum _{i=1}^{s-1}\left( \sum _{m=2}^\infty \frac{\vert x_{n_{i+1}}(m)-x_{n_i}(m)\vert }{2^{m-1}}\right) -\sum _{m=2}^\infty \frac{\vert x_{n_1}(m)-x_{n_s}(m)\vert }{2^{m-1}}. \end{aligned}$$

By the triangle inequality:

$$\begin{aligned} \sum _{m=2}^\infty \frac{1}{2^{m-1}}\left( \sum _{i=1}^{s-1}\vert x_{n_{i+1}}(m)-x_{n_i}(m)\vert -\vert x_{n_1}(m)-x_{n_s}(m)\vert \right) \ge 0. \end{aligned}$$

From this:

$$\begin{aligned} \sum _{i=1}^{s-1} \Vert x_{n_{i+1}}-x_{n_i}\Vert _\infty -\Vert x_{n_1}-x_{n_s}\Vert _\infty \le b \end{aligned}$$

and $(x_n)$ is a directional sequence in $(c_0,\Vert \cdot \Vert _\infty )$, so if C is directionally bounded in $(c_0,\Vert \cdot \Vert _\infty )$, it is in $(c_0,\vert \Vert \cdot \Vert \vert )$. $\square$

Finally, from the above theorem, we conclude that there exist two equivalent renormings of a Banach space X whose Banach-Mazur distance is 1 and they share the collection of sets with the AFPP.

Corollary 1

A closed convex set $C\subset c_0$ has the AFPP in $(c_0,\Vert \cdot \Vert )$ if and only if it has the AFPP in $(c_0,\vert \Vert \cdot \Vert \vert )$. Moreover, $d((c_0,\Vert \cdot \Vert ),(c_0,\vert \Vert \cdot \Vert \vert ))=1$.

Proof

Theorem 11 in [4] implies that C is directionally bounded in $(c_0,\Vert \cdot \Vert )$ if and only if C is directionally bounded in $c_0$. The conclusion follows from Theorems 3 and 10. The last statement was proved in Lemma 3. $\square$