Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

Abstract

We show that each infinite-dimensional reflexive Banach space $(X,\left\| \cdot \right\| _X)$ has an equivalent norm $\left\| \cdot \right\| _{X,0}$ such that $(X,\left\| \cdot \right\| _{X,0})$ is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space $C([0,1],{\mathbb {R}})$ contains a constant width set with empty interior.

1 Introduction

In this paper we study the following problem: when does an infinite-dimensional Banach space $(X,\left\| \cdot \right\| _X)$ have an equivalent norm $\left\| \cdot \right\| _{X,0}$ such that the Banach space $(X,\left\| \cdot \right\| _{X,0})$ contains a diametrically complete set with empty interior?

Thus our paper is a complement to the papers [7, 24] and [39]. More precisely, in [7] it is proved that such a norm exists for each infinite-dimensional, separable and reflexive Banach space. In [24] we extend this result and prove that for each infinite-dimensional reflexive Banach space $(X,\left\| \cdot \right\| _X)$ with the nonstrict Opial and the Kadec-Klee properties, there exists an equivalent norm such that X equipped with this norm is LUR and contains a diametrically complete set with empty interior. These additional assumptions on the Banach space $(X,\left\| \cdot \right\| _X)$ indicate the limit of applications of the Maluta method in proving results of this type ([36]) and therefore in the nonseparable case we need a new approach to obtain our main theorem (see Sect. 3).

In [39] E. Maluta and D. Yost prove that each Banach space $(X,\left\| \cdot \right\| _X)$ with an unconditional Schauder basis can be renormed in such a way that X with this new norm contains a constant width set with empty interior. In Sect. 3 we extend this result and using it in the next section, we show that after a suitable equivalent renorming the Banach space $C([0,1],{\mathbb {R}})$ contains a constant width set with empty interior. In Sect. 4 we also compare and contrast our results. Moreover, in the Appendix, we improve Theorem 4.13 of [7] which is one of the crucial tools in the proof of the main theorem in that paper.

2 Basic Notions and Facts

Throughout this paper all Banach spaces are over the field of reals. We use the notations and notions given in [24]. In the sequel we also use the standard definitions of unconditional series, unconditional basic sequence, Schauder basis and unconditional Schauder basis. The known properties of these notions in an infinite-dimensional Banach space can be found in [1, 21, 33, 45,46,47] and [48].

In this section we recall several additional definitions, notions and facts regarding the geometry of Banach spaces.

We begin with the definition of a complemented subspace.

Definition 2.1

([23]) Let $(X,\left\| \cdot \right\| _X)$ be a Banach space and let $Y \ne X$ be a closed subspace of X with $\textrm{dim} \, Y \ge 1$. If there exists a continuous linear projection of X onto Y, then we say that the subspace Y is complemented in X.

We have the following theorems which are closely connected with this notion.

Theorem 2.1

([4, 50, 53]) Let $(X,\left\| \cdot \right\| _X)$ be a separable Banach space. If there exists a closed subspace Y of X which is isomorphic to $c_0$, then Y contains a closed subspace $Y_1$ isomorphic to $c_0$, which is complemented in X.

Theorem 2.2

([1, 3]) $C([0,1], {\mathbb {R}})$ with the standard max-norm $\left\| \cdot \right\| _C$ has a closed subspace which is isometric to $c_0$.

However, observe that in the case of $C([0,1], {\mathbb {R}})$ we also have the following negative result.

Theorem 2.3

([18]) No reflexive infinite-dimensional subspace of $C([0,1], {\mathbb {R}})$ with the standard max-norm has a complement.

The next result regarding existence of a separable complemented subspace in a nonseparable Banach space is due to J. Lindenstrauss. We present it in a weaker form which is suitable for our subsequent considerations.

Theorem 2.4

([32]). Let $(X,\left\| \cdot \right\| _X)$ be a nonseparable and reflexive Banach space. Then there exists a linear, separable and infinite-dimensional subspace $X_1$ of X, and a linear projection P of X onto $X_1$ with the operator norm $\Vert P\Vert$ equal to 1.

Now, we recall the notion of uniform convexity in every direction.

Definition 2.2

([14], see also [54]). For a Banach space $(X,\left\| \cdot \right\| _X)$ and a fixed element $z \in X$ with $\Vert z\Vert _X=1$, the function $\delta _z:[0,2] \rightarrow [0,1]$ defined by

$$\begin{aligned} \delta _z(\epsilon ) := \inf \{1-\frac{1}{2} \Vert x+y\Vert _X: \Vert x\Vert _X \le 1, \; \Vert y\Vert _X \le 1, \; x -y = \epsilon z\} \end{aligned}$$

is called the modulus of convexity of $(X,\left\| \cdot \right\| _X)$ in the direction z.

If $\delta _z(\epsilon ) >0$ for all $\epsilon >0$, then $(X,\left\| \cdot \right\| _X)$ is said to be uniformly convex in the direction z.

If $\delta _z(\epsilon ) >0$ for all $\epsilon >0$ and all $z \in X$ with $\Vert z\Vert =1$ , then $(X,\left\| \cdot \right\| _X)$ is said to be uniformly convex in every direction.

In [54] V. Zizler proved the following two results.

Proposition 2.1

Suppose that $(X,\left\| \cdot \right\| _X)$ and $(Y,\left\| \cdot \right\| _Y)$ are Banach spaces, T is a linear continuous one-to-one mapping of X into Y. Assume that $(Y,\left\| \cdot \right\| _Y)$ is uniformly convex in every direction $Tz/\Vert Tz\Vert _Y$, $z \in X$ and $\Vert z\Vert _X =1$. Then X has an equivalent norm given by

for $x \in X$, which is uniformly convex in every direction.

Corollary 2.1

Assume that a Banach space $(X,\left\| \cdot \right\| _X)$ has a bounded sequence of functionals $\{f_i^*\}_{i=1}^\infty$ in $(X^*,\left\| \cdot \right\| _{X^*})$ which separates the points in $(X,\left\| \cdot \right\| _X)$, $\beta > 0$ and that an equivalent norm in X is given by

for $x \in X$. Then is uniformly convex in every direction.

Next, we recall some results about locally uniformly rotund Banach spaces and Banach spaces with the Kadec-Klee property.

Definition 2.3

([35]). We say that a Banach space $(X,\left\| \cdot \right\| _X)$ is locally uniformly rotund (LUR) if for each $x \in X$, every sequence $\{x_k\}_{k=1}^\infty$ with $\lim _k\Vert x_k\Vert _X =\Vert x\Vert _X$ and $\lim _k\Vert x+x_k\Vert _X=2\Vert x\Vert _X$ tends strongly to x. In this case we also say that the norm $\left\| \cdot \right\| _X$ is LUR.

The next definition is equivalent to the above definition of a locally uniformly rotund Banach space.

Definition 2.4

([54]). We say that a Banach space $(X,\left\| \cdot \right\| _X)$ is locally uniformly rotund (LUR) if for each $x \in X$, every sequence $\{x_k\}_{k=1}^\infty$ satisfying

$$\begin{aligned} \lim _k\left[ 2 \left( \Vert x\Vert _X^2 +\Vert x_k\Vert _X^2\right) -\Vert x+x_k\Vert _X^2\right] =0, \end{aligned}$$

tends strongly to x.

Definition 2.5

([25, 26] and [30]; see also [15]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space. We say that $(X,\left\| \cdot \right\| _X)$ has the Kadec-Klee property with respect to the weak topology (the Kadec-Klee property for short) if each sequence $\{x_k\}_{k=1}^\infty$ with $\lim _k\Vert x_k\Vert _X =1$, which converges weakly to a point x with $\Vert x\Vert _X =1$, tends strongly to x.

The following theorem provides a connection between these two notions.

Theorem 2.5

([10]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space. If $(X,\left\| \cdot \right\| _X)$ is locally uniformly rotund, then $(X,\left\| \cdot \right\| _X)$ has the Kadec-Klee property with respect to the weak topology.

M. I. Kadec proved the following theorem concerning a renorming of a separable Banach space.

Theorem 2.6

([27], see also [10] and [13]). Let $(X,\left\| \cdot \right\| _X)$ be a separable Banach space, $\{X_i\}_{i=1}^\infty$ an increasing sequence of finite dimensional subspaces of X such that $\bigcup _i X_i$ is dense in $(X,\left\| \cdot \right\| _X)$, $\{f_i^*\}_{i=1}^\infty$ a separating (for X) family of functionals with $\Vert f_i^*\Vert _{X^*} =1$ for each $i \in {\mathbb {N}}$, and let $\beta _1$, $\beta _2$ be positive real numbers. Define a new norm on X by

for $x \in X$. Then is an equivalent norm to $\left\| \cdot \right\| _X$ on X,

and is LUR.

The following result is due to A. R. Lovaglia.

Theorem 2.7

([35]). Let the Banach spaces $(X_1,\left\| \cdot \right\| _{X_1})$ and $(X_2,\left\| \cdot \right\| _{X_2})$ be locally uniformly rotund (LUR). If on $X:=X_1\times X_2$ the norm $\left\| \cdot \right\| _X$ is defined by

$$\begin{aligned} \Vert x\Vert _X=\Vert (x^1,x^2)\Vert := \sqrt{\Vert x^1\Vert _{X_1}^2 + \Vert x^2\Vert _{X_2}^2} \end{aligned}$$

for $x=(x^1,x^2) \in X$, then $(X,\left\| \cdot \right\| _X)$ is also LUR.

The next theorem is a deep result which is due to S. L. Troyanski.

Theorem 2.8

([51]). Every reflexive Banach space admits an equivalent locally uniformly rotund norm.

Now, we recall the notions of Opial property and nonstrict Opial property.

Definition 2.6

([29, 43]). A Banach space $(X,\left\| \cdot \right\| _X)$ is said to have the Opial property if for each weakly null sequence $\{x_k\}_{k=1}^\infty$ and each $x \ne 0$ in X, we have

$$\begin{aligned} \limsup _{k}\Vert x_k\Vert _X < \limsup _{k}\Vert x_k - x\Vert _X. \end{aligned}$$

A Banach space $(X,\left\| \cdot \right\| _X)$ is said to have the nonstrict Opial property if for each weakly null sequence $\{x_k\}_{k=1}^\infty$ and each point x in X, we have

$$\begin{aligned} \limsup _{k}\Vert x_k\Vert _X \le \limsup _{k}\Vert x_k - x\Vert _X. \end{aligned}$$

Remark 2.1

In [43] Z. Opial showed that for $1<p<\infty$ and $p\ne 2$, the Banach space $(L^p([0,1],{\mathbb {R}})$ with the standard norm does not have the nonstrict Opial property.

Theorem 2.9

([52]). (1) Assume that a Banach space $(X,\left\| \cdot \right\| _X)$ has the nonstrict Opial property and that $\beta$ is a positive real number. If a bounded sequence of functionals $\{f_i^*\}_{i=1}^\infty$ in $(X^*,\left\| \cdot \right\| _{X^*})$ separates the points in $(X,\left\| \cdot \right\| _X)$ and if an equivalent norm on X is given by

for $x \in X$, then has the Opial property.

(2) There exists a universal constant ${\check{K}}\ge 1$ such that every infinite-dimensional and separable Banach space $(X,\left\| \cdot \right\| _X)$ admits an equivalent norm $\left\| \cdot \right\| _{X,1}$ with the Opial property, and

In addition, is uniformly convex in every direction.

Here we have to recall the following theorem which is motivated by the proof of van Dulst’s result ([52]).

Theorem 2.10

([6]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space with a Schauder basis $\{e_i\}_i$ and let ${\mathcal {P}}=\{P_n\}_{n=0}^\infty$ be the sequence of the natural projections associated with this basis, that is, $P_0:=0$ and $P_n(x):=P_n(\sum _{i=1}^\infty x^i e_i):= \sum _{i=1}^n x^ie_i$ for each $x =\sum _{i=1}^\infty x^i e_i \in X$. Then the norm $\left\| \cdot \right\| _{{\mathcal {P}},X}$ defined on X by

$$\begin{aligned} \Vert x\Vert _{{\mathcal {P}},X}= \sup _{n=0,1,...} \Vert x -P_n x\Vert _X \end{aligned}$$

for each $x \in X$ is equivalent to the norm $\left\| \cdot \right\| _X$ and the Banach space $(X,\left\| \cdot \right\| _{{\mathcal {P}},X})$ has the nonstrict Opial property.

The next three definitions are closely connected with the concept of the diameter of a set.

Definition 2.7

([40]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space and let C be a nonempty, bounded and non-singleton subset of X. We say that C is a diametrically complete set in X if

$$\begin{aligned} {\textrm{diam}}_{\left\| \cdot \right\| _X}(C \cup \{x\}) > {\textrm{diam}}_{\left\| \cdot \right\| _X}(C) \end{aligned}$$

for each $x \in X \setminus C$.

It is obvious that a diametrically complete set has to be closed and convex. Many other interesting properties of this kind of sets can be found in [11, 36,37,38,39, 41] and in the literature given therein.

Definition 2.8

([5]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space. For a nonempty, bounded and convex set $C \subset X$, the number

$$\begin{aligned} r_{\left\| \cdot \right\| _X}(C,C) := \inf \{\sup \{\Vert x - x'\Vert _X: x' \in C\}: x \in C\} \end{aligned}$$

is called the Chebyshev self-radius of C.

Definition 2.9

([5]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space and let C be a nonempty, bounded and convex subset of X. We say that the set C is diametral if $r_{\left\| \cdot \right\| _X}(C,C) = {\textrm{diam}}_{\left\| \cdot \right\| _X}(C)$. If $(X,\left\| \cdot \right\| _X)$ does not contain any nonempty, bounded, convex, diametral and non-singleton subset, then we say that $(X,\left\| \cdot \right\| _X)$ has normal structure.

The following theorem, which is due to J. P. Moreno, P. L. Papini and R. R. Phelps, exhibits a connection between the diametral property of a set and the emptiness of the interior of a diametrically complete set.

Theorem 2.11

([42]). Let $(X,\left\| \cdot \right\| _X)$ be an infinite-dimensional Banach space and let a set $C\subset X$ be diametrically complete. If the interior of C is empty, then C is diametral.

E. Maluta and P. L. Papini proved the following result, which provides a sufficient condition for the existence of diametrically complete sets the interior of which is empty.

Theorem 2.12

([38]). Each reflexive Banach space $(X,\left\| \cdot \right\| _X)$, which has the nonstrict Opial property and lacks normal structure, contains diametrically complete sets the interior of which is empty.

Now, we recall another result, which motivates our investigations in the next section.

Theorem 2.13

([7]). Each infinite-dimensional reflexive and separable Banach space $(X,\left\| \cdot \right\| _X)$ has an equivalent norm $\left\| \cdot \right\| _{X,0}$ such that $(X,\left\| \cdot \right\| _{X,0})$ is LUR and contains a diametrically complete set the interior of which is empty.

In Sect. 3 we also consider constant width sets.

Definition 2.10

([19, 31]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space. A bounded, closed, convex and non-singleton set $C \subset X$ is said to be of constant width $\lambda$ (or, more simply, of constant width) if for every $f \in X^*$, $\Vert f\Vert _{X^*} =1$, we have

$$\begin{aligned} \sup f(C) - \inf f(C) = \sup f(C-C) = \lambda . \end{aligned}$$

Note that if C is of constant width $\lambda$, then necessarily $\lambda = {\textrm{diam}}_{\left\| \cdot \right\| _X}(C)>0$.

The notion of a diametrically complete set is closely connected with the notion of a constant width set. It is generally known that all closed balls are both of constant width and diametrically complete sets. These notions are also equivalent in two-dimensional Banach spaces, in all finite-dimensional Euclidean spaces and in Hilbert spaces. Moreover, every set of constant width is diametrically complete ([40]). On the other hand, relying on the 3-dimensional result due to Eggleston ([12]), Moreno et al. ([42]) have shown that if X is any Banach space of dimension at least 3, then (after a suitable equivalent renorming) it contains a set which has nonempty interior and is diametrically complete, but is not of constant width.

We have the following characterization of constant width sets.

Theorem 2.14

([44]). Let $(X,\left\| \cdot \right\| _X)$ be a Banach space. Suppose that $C \subset X$ is a bounded, closed and convex set of positive diameter $\lambda = \textrm{diam}_{\left\| \cdot \right\| _X}$. Then the following statements are equivalent:

• (a) C is of constant width $\lambda$.

• (b) $C-C$ contains the interior of the ball $\lambda \overline{B_X(0,1)}$.

• (c) $C-C$ is dense in the ball $\lambda \overline{B_X(0,1)}$.

E. Maluta and D. Yost have recently proved the following theorem.

Theorem 2.15

([39]). Every infinite-dimensional Banach space $(X,\left\| \cdot \right\| _X)$ which admits an unconditional Schauder basis has an equivalent norm $\left\| \cdot \right\| _{X,0}$ such that $(X,\left\| \cdot \right\| _{X,0})$ contains a constant width set with empty interior.

3 Diametrically Complete Sets and Constant Width Sets

In this section we proceed as follows. First, by successively applying the Lindenstrauss theorem, our auxiliary theorem (Theorem 3.1 below) and the Troyanski theorem, the problem of existence of an equivalent norm $\left\| \cdot \right\| _{X,0}$ in any reflexive infinite-dimensional Banach space $(X,\left\| \cdot \right\| _X)$ such that the Banach space $(X,\left\| \cdot \right\| _{X,0})$ contains a diametrically complete set with empty interior is reduced to the separable case. Next, we simply use the main result of [7] (see Theorem 2.13 above) and the Lovaglia theorem to obtain the claimed LUR norm. Finally, we generalize the Maluta-Yost result.

We begin with the above-mentioned auxiliary theorem.

Theorem 3.1

Let $(X_1,\left\| \cdot \right\| _{X_1})$ and $(X_2,\left\| \cdot \right\| _{X_2})$ be Banach spaces.

(a) Let $X=X_1 \times X_2$ be endowed with the norm

$$\begin{aligned} \Vert x\Vert _X:= \sqrt{\Vert x^1\Vert _{X_1}^2 + \Vert x^2\Vert _{X_2}^2}, \end{aligned}$$

where $x = (x^1,x^2) \in X.$ If $(X_1,\left\| \cdot \right\| _{X_1})$ contains a diametrically complete set the interior of which is empty, then $(X,\left\| \cdot \right\| _X)$ also contains a diametrically complete set the interior of which is empty.

(b) Suppose $X=X_1 \times X_2$ has the norm

$$\begin{aligned} \Vert x\Vert _X:= \max \{\Vert x^1\Vert _{X_1}, \Vert x^2\Vert _{X_2}\}, \end{aligned}$$

where $x = (x^1,x^2) \in X.$ If $C_1 \subset X_1$ and $C_2 \subset X_2$ are constant width (diametrically complete) sets in $(X_1,\left\| \cdot \right\| _{X_1})$ and $(X_2,\left\| \cdot \right\| _{X_2})$, respectively, and $\textrm{diam}_{\left\| \cdot \right\| _{X_1}} \;C_1=\textrm{diam}_{\left\| \cdot \right\| _{X_2}} \;C_2=\lambda$, then $C=C_1\times C_2$ is also a constant width (diametrically complete) set in $(X,\left\| \cdot \right\| _X)$ with $\textrm{diam}_{\left\| \cdot \right\| _X} \;C=\lambda$.

(c) Let $X=X_1 \times X_2$ be endowed with the norm

$$\begin{aligned} \Vert x\Vert _X:= \max \{\Vert x^1\Vert _{X_1}, \Vert x^2\Vert _{X_2}\}, \end{aligned}$$

where $x = (x^1,x^2) \in X.$ Assume that $(X_1,\left\| \cdot \right\| _{X_1})$ contains a constant width set with empty interior. Then $(X,\left\| \cdot \right\| _X)$ also contains a constant width set with empty interior.

Proof

(a) Let $C_1 \subset X_1$ be a diametrically complete set in $(X_1,\left\| \cdot \right\| _{X_1})$ such that the interior of $C_1$ in $(X_1,\left\| \cdot \right\| _{X_1})$ is empty. Set $C := C_1 \times \{0\} \subset X$. It is obvious that the interior of C is empty in $(X,\left\| \cdot \right\| _X)$ and that ${\textrm{diam}}_{\left\| \cdot \right\| _X} C = {\textrm{diam}}_{\left\| \cdot \right\| _{X_1}} C_1$. To show that C is diametrically complete, we take $x=(x^1,x^2) \in X$, $x \notin C$, and consider the following two cases.

Case 1. $x^1 \notin C_1$. Since $C_1$ is diametrically complete in $(X_1,\left\| \cdot \right\| _{X_1})$ we have

$$\begin{aligned}{} & {} {\textrm{diam}}_{\left\| \cdot \right\| _X}(C \cup \{x\})\ge \sup \{\Vert ({\tilde{x}}^1,0) - (x^1,x^2)\Vert _{X}: ({\tilde{x}}^1,0) \in C\} \\{} & {} \quad = \sup \{ \sqrt{\Vert {\tilde{x}}^1-x^1\Vert _{X_1}^2 + \Vert x^2\Vert _{X_2}^2}: {\tilde{x}}^1 \in C_1\} \ge \sup \{\Vert {\tilde{x}}^1 - x^1\Vert _{X_1}: {\tilde{x}}^1 \in C_1\} \\{} & {} \quad > {\textrm{diam}}_{\left\| \cdot \right\| _{X_1}} C_1 = {\textrm{diam}}_{\left\| \cdot \right\| _X }C. \end{aligned}$$

Case 2. $x^1 \in C_1$. Since $x^2 \ne 0$ and $C_1$ is diametral (see Theorem 2.11), we get

$$\begin{aligned}{} & {} {\textrm{diam}}_{\left\| \cdot \right\| _X}(C \cup \{x\}) \ge \sup \{ \sqrt{\Vert {\tilde{x}}^1-x^1\Vert _{X_1}^2 + \Vert x^2\Vert _{X_2}^2}: {\tilde{x}}^1 \in C_1\} \\{} & {} \quad > \sup \{\Vert {\tilde{x}}^1-x^1\Vert _{X_1}: {\tilde{x}}^1 \in C_1\} = {\textrm{diam}}_{\left\| \cdot \right\| _{X_1}} C_1 = {\textrm{diam}}_{\left\| \cdot \right\| _X} C. \end{aligned}$$

(b) In the case of constant width sets this is a direct consequence of, for example, Theorem 2.14.

Now, let $C_1 \subset X_1$ and $C_2 \subset X_2$ be diametrically complete sets. To show that C is diametrically complete, we take $x=(x^1,x^2) \in X$, $x \notin C$. Without loss of generality we may assume that $x^1 \notin C_1$. Since $C_1$ is diametrically complete in $(X_1,\left\| \cdot \right\| _{X_1})$, we have

$$\begin{aligned}{} & {} {\textrm{diam}}_{\left\| \cdot \right\| _X}(C \cup \{x\})\ge \sup \{\max \{\Vert {\tilde{x}}^1 - x^1\Vert _{X_1}, \Vert {\tilde{x}}^2 - x^2\Vert _{X_2}\}: ({\tilde{x}}^1,{\tilde{x}}^2) \in C\} \\{} & {} \quad \ge \sup \{\Vert {\tilde{x}}^1 - x^1\Vert _{X_1}: {\tilde{x}}^1 \in C_1\} > {\textrm{diam}}_{\left\| \cdot \right\| _{X_1}} C_1 = \lambda = {\textrm{diam}}_{\left\| \cdot \right\| _X }C. \end{aligned}$$

(c) Let $C_1 \subset X_1$ be a constant width set in $(X_1,\left\| \cdot \right\| _{x_1})$ such that the interior of $C_1$ in $(X_1,\left\| \cdot \right\| _{X_1})$ is empty. Without any loss of generality we may assume that $\textrm{diam}_{\left\| \cdot \right\| _{X_1}} \;C_1 =2.$ Set $C := C_1 \times \overline{B_{X_2}(0,1)} \subset X$, where $\overline{B_{X_2}(0,1)}$ is the closed unit ball in $(X_2,\left\| \cdot \right\| _{X_2})$. By (b) the set C is a constant width set in $(X,\left\| \cdot \right\| _X)$. It is obvious that the interior of C is empty in $(X,\left\| \cdot \right\| _X)$. This completes the proof. $\square$

Combining Theorem 3.1 with Theorems 2.4, 2.7, 2.8 and 2.13 we can now prove our basic theorem on diametrically complete sets with empty interior.

Theorem 3.2

Each infinite-dimensional reflexive Banach space $(X,\left\| \cdot \right\| _X)$ has an equivalent norm $\left\| \cdot \right\| _{X,0}$ such that $(X,\left\| \cdot \right\| _{X,0})$ is LUR and contains a diametrically complete set the interior of which is empty.

Proof

In view of Theorem 2.13, it suffices to consider a nonseparable Banach space $(X,\left\| \cdot \right\| _X)$. Then by Theorem 2.4 there exists a linear and separable subspace $X_1$ of X, and a linear projection P of X onto $X_1$ with its operator norm $\Vert P\Vert$ equal to 1. Hence $X:=X_1 \oplus X_2$, where $X_2:=(I-P)(X)$ and I is the identity operator on X. It then follows from Theorem 2.13 that $(X_1,\left\| \cdot \right\| _X)$ has an equivalent norm $\left\| \cdot \right\| _{X_1,0}$ such that $(X_1,\left\| \cdot \right\| _{X_1,0})$ is LUR and contains a diametrically complete set C the interior of which is empty. Moreover, by Troyanski’s theorem (Theorem 2.8), the Banach space $(X_2,\left\| \cdot \right\| _{X})$ admits an equivalent locally uniformly rotund norm $\left\| \cdot \right\| _{X_2,1}$. So we can define on X an equivalent norm $\left\| \cdot \right\| _{X,0}$ by setting

$$\begin{aligned} \Vert x\Vert _{X,0}:= \sqrt{\Vert x^1\Vert _{X_1,0}^2 + \Vert x^2\Vert _{X_2,1}^2} \end{aligned}$$

for $x = x^1 +x^2 \in X$, where $x^1:=Px \in X_1$ and $x^2 :=(I-P)x \in X_2$. Theorem 2.7 implies that $(X,\left\| \cdot \right\| _{X,0})$ is LUR. To prove that $C \subset X_1 \subset X$ is a diametrically complete set the interior of which is empty, it is now sufficient to apply Theorem 3.1. $\square$

As a corollary of Theorems 3.1 and 3.2 we get the following generalization of Theorem 3.2.

Theorem 3.3

Let $(X_1,\left\| \cdot \right\| _{X_1})$ and $(X_2,\left\| \cdot \right\| _{X_2})$ be Banach spaces, and let $X=X_1 \times X_2$ be endowed with the norm

$$\begin{aligned} \Vert x\Vert _X:= \sqrt{\Vert x^1\Vert _{X_1}^2 + \Vert x^2\Vert _{X_2}^2}, \end{aligned}$$

where $x = (x^1,x^2) \in X.$ If $(X_1,\left\| \cdot \right\| _{X_1})$ is infinite-dimensional and reflexive, then $(X,\left\| \cdot \right\| _X)$ has an equivalent renorming $(X,\left\| \cdot \right\| _{X,0})$ that contains a diametrically complete set the interior of which is empty.

Similarly, Theorems 2.15 and 3.1 imply the following generalization of the Maluta-Yost result.

Theorem 3.4

Let $(X_1,\left\| \cdot \right\| _{X_1})$ and $(X_2,\left\| \cdot \right\| _{X_2})$ be infinite-dimensional Banach spaces, and let $X=X_1 \times X_2$ be endowed with the norm

$$\begin{aligned} \Vert x\Vert _X:= \max \{\Vert x^1\Vert _{X_1}, \Vert x^2\Vert _{X_2}\}, \end{aligned}$$

where $x = (x^1,x^2) \in X.$ If $(X_1,\left\| \cdot \right\| _{X_1})$ admits an unconditional Schauder basis, then $(X,\left\| \cdot \right\| _X)$ has an equivalent renorming $(X,\left\| \cdot \right\| _{X,0})$ that contains a constant width set with empty interior.

4 Some Applications of Our Results

First we want to extend the Maluta-Yost theorem to nonseparable Banach spaces. Therefore we have to recall that in 1939 E. B. Lorch ([34]) introduced the concept of an unconditional extended basis in a nonseparable Banach space $(X,\left\| \cdot \right\| _X)$.

Definition 4.1

([34, 48]). Let $(X,\left\| \cdot \right\| _X)$ be a nonseparable Banach space. An uncountable family $\{x_i\}_{i \in I}$ of elements of X is called an unconditional extended basis in $(X,\left\| \cdot \right\| _X)$ if $\overline{\textrm{span}}\{x_i\}_{i \in I}= X$ and if every countable subfamily of $\{x_i\}_{i \in I}$ is an unconditional basic sequence. In this case, for each $x \in X$, we write $x=\sum _{i \in I} a^i x_i$, where the family $\{a^i\}_{i\in I}$ of coefficients is uniquely determined by this unconditional extended basis and by x.

The theory of unconditional extended bases can be found, for example, in [20] and [48].

Let us consider such a Banach space. Observe that if $(X,\left\| \cdot \right\| _X)$ admits an unconditional extended basis $\{x_i\}_{i \in I}$, then, without loss of generality, we can set $I = {\mathbb {N}} \cup I_1$, where ${\mathbb {N}} \cap I_1 = \emptyset$. Therefore we have

$$\begin{aligned} X= \overline{\textrm{span}} \{x_i\}_{i \in {\mathbb {N}}} \oplus \overline{\textrm{span}} \{x_i\}_{i \in I_1}. \end{aligned}$$

This means that Theorem 3.4 can be applied suitably to any space X with an unconditional extended basis and every space $X=X_1 \times X_2$ in which $X_1$ has an unconditional extended basis.

Since the Banach space $C([0,1], {\mathbb {R}})$ with the standard max-norm $\left\| \cdot \right\| _C$ is not isomorphic to a subspace of a Banach space with an unconditional Schauder basis ([28, 33]) and no reflexive infinite-dimensional subspace of $(C([0,1], {\mathbb {R}}, \Vert \cdot \Vert _C))$ has a complement (Theorem 2.3), it is obvious that Theorems 2.13, 2.15 do not apply to the case of the Banach space $C([0,1], {\mathbb {R}})$. We are, however, able to show how to apply Theorem 3.4 to this space.

Theorem 4.1

The Banach space $C([0,1], {\mathbb {R}})$ with the standard max-norm $\left\| \cdot \right\| _C$ has an equivalent norm $\left\| \cdot \right\| _{C,0}$ such that $(C([0,1], {\mathbb {R}}),\left\| \cdot \right\| _0)$ contains a constant width set the interior of which is empty.

Proof

By Theorems 2.1 and 2.2, the Banach space $(C([0,1], {\mathbb {R}}),\left\| \cdot \right\| _C)$ contains a complemented copy of $c_0$ and therefore Theorem 3.4 implies the claimed result. $\square$

Remark 4.1

It is worthwhile observing that in $C([0,1],{\mathbb {R}})$ with the max-norm, the only constant width sets are closed balls, but there do exist diametrically complete sets with empty interior [42].

Now, we give a few examples which seem to be interesting from the point of view of the theory of diametrically complete sets. First, observe also that the condition given in Theorem 2.12 ([38]) for the existence of diametrically complete sets the interior of which is empty is only sufficient, but not necessary, as the following example shows.

Example 4.1

In [36] E. Maluta has shown that ${\ell }^2$ furnished with the Day type norm $\left\| \cdot \right\| _{L}$ ([49]) is LUR and contains diametrically complete sets with empty interior. We consider, in addition, the Banach space $(L^p([0,1], {\mathbb {R}})$, with the standard norm $\left\| \cdot \right\| _p$, $1<p<\infty$, $p \ne 2$, and we set

$$\begin{aligned} X := {\ell }^2 \times L^p([0,1], {\mathbb {R}}) \end{aligned}$$

with the norm

The reflexive and separable Banach space is LUR (see Theorem 2.7) and does not have the nonstrict Opial property (see Remark 2.1), but by Theorem 3.1 it does contain diametrically complete sets the interior of which is empty.

Next, we present an interesting and pertinent example of a certain nonreflexive Banach space.

Example 4.2

It is not known whether any reflexive Banach space $(X, \left\| \cdot \right\| _X)$ can be renormed so as to have the nonstrict Opial property. This statement is false in general as the Banach space ${\ell }^\infty (\Gamma )$ with an uncountable $\Gamma$ shows ([8]). In addition, ${\ell }^\infty (\Gamma )$ with an uncountable $\Gamma$ cannot be equivalently renormed so as to be LUR or even strictly convex ([9]). Hence for uncountable $\Gamma$, the space

$$\begin{aligned} X :={\ell }^2 \times {\ell }^\infty (\Gamma ) \end{aligned}$$

with the norm

(where $\left\| \cdot \right\| _\infty$ is the standard sup-norm on ${\ell }^\infty (\Gamma )$) admits no norm with the nonstrict Opial property and no strictly convex norm, but by E. Maluta’s result ([36]) and Theorem 3.1 it does contain diametrically complete sets the interior of which is empty.

We can also present a construction of Banach spaces to which we can apply Theorem 3.4.

Example 4.3

Let the set $\Gamma$ be at least countable, $\{(Y_\gamma , \left\| \cdot \right\| _\gamma \}_{\gamma \in \Gamma }$ be a family of Banach spaces with $\mathrm {\dim }Y_\gamma \ge 2$ and let $1\le p <\infty$. Set

$$\begin{aligned} X = {\ell }^p(\{Y_\gamma \}_\Gamma ) := \{x=\{y_\gamma \}_{\gamma \in \Gamma } \in \prod _{\gamma \in \Gamma } Y_\gamma : \sum _{\gamma \in \Gamma } \Vert y_\gamma \Vert _\gamma ^p < \infty \} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert x\Vert _{X} := \left[ \sum _{\gamma \in \Gamma } \Vert y_\gamma \Vert ^p\right] ^{\frac{1}{p}}. \end{aligned}$$

Since each $Y_\gamma$ can be represented as

$$\begin{aligned} Y_\gamma :={\mathbb {R}} \oplus Y_{\gamma ,1} = P_{{\mathbb {R}}, \gamma }(Y) \oplus (I - P_{{\mathbb {R}}, \gamma }(Y)), \end{aligned}$$

(where $\Vert P_{{\mathbb {R}}, \gamma }\Vert =1$) with the equivalent norm

$$\begin{aligned} \Vert y_\gamma \Vert _{\gamma ,1}=\Vert (t,y_{\gamma ,1})\Vert _{\gamma ,1}:= \root p \of {t^p + \Vert y_{\gamma ,1}\Vert _{\gamma }^p} \end{aligned}$$

for $y_\gamma =(t,y_{\gamma ,1}) \in Y_\gamma$, we have $X = {\ell }^p(\{Y_\gamma \}_{\gamma \in \Gamma })={\ell }^p(\Gamma ) \oplus {\ell }^p(\{Y_{\gamma ,1}\}_{\gamma \in \Gamma })$.

It is not difficult to observe that directly from the definition of constant width sets in a Banach space $(X, \left\| \cdot \right\| _X)$ it follows that

$$\begin{aligned} \overline{\textrm{span}\; C}=X \end{aligned}$$

for each constant width set $C \subset X$. Hence the following theorem is a simple consequence of Theorems 3.1 and 3.2.

Theorem 4.2

On every infinite-dimensional and reflexive Banach space $(X, \left\| \cdot \right\| _X)$ there exists an equivalent norm such that X with this new norm contains diametrically complete sets with empty interior which are not constant width sets.

Proof

Indeed, consider an arbitrary subspace $X_1$ of X with $\textrm{codim} X_1 =1$. Then we have $X= X_1 \oplus {\mathbb {R}}$. By reflexivity of $(X, \left\| \cdot \right\| _X)$ and Theorem 3.2, $X_1$ has a norm $\left\| \cdot \right\| _{X_1,0}$, which is equivalent to the norm $\left\| \cdot \right\| _X$ restricted to $X_1$, such that the Banach space $(X_1, \left\| \cdot \right\| _{X_1,0})$ contains a diametrically complete set $C_1$ with empty interior (see also Theorem 3.8 in [38]). Therefore it suffices to take the norm $\left\| \cdot \right\| _{X,0}$ on X defined by

$$\begin{aligned} \Vert (x^1,t)\Vert _{X,0}:= \sqrt{\Vert x^1\Vert _{X_1,0}^2 + t^2} \end{aligned}$$

and then let $C:=C_1 \times \{0\}$ (see the proof of Theorem 3.1 (a)). $\square$

Remark 4.2

We get similar results when in Theorem 4.2 we replace the assumption of reflexivity of the Banach space $(X, \left\| \cdot \right\| )$ by the assumptions which appear either in Theorem 2.15 or in Theorem 3.3 or in Theorem 3.4.

Remark 4.3

We have some restrictions in applications of Theorems 2.15, 3.2, 3.3 and 3.4. More precisely, the following facts are valid. There exists an infinite-dimensional and reflexive Banach space which does not contain unconditional basic sequences ([17]) and therefore this Banach space satisfies the assumptions of Theorem 3.2, but it does not satisfy the assumptions of Theorems 2.15 and 3.4. Next, there exists a nonreflexive Banach space which contains neither an unconditional basic sequence nor does it contain an infinite-dimensional reflexive subspace ([16], see also [2]) and this implies the existence of Banach spaces which do not satisfy the assumptions of Theorems 2.15, 3.2, 3.3 and 3.4.

Appendix: Construction of a Norm Which is LUR and has the Opial Property

The following theorem is one of the basic tools in [7].

Theorem 5.1

Every infinite-dimensional and separable Banach space $(X,\left\| \cdot \right\| _X)$ admits an equivalent norm $\left\| \cdot \right\| _{X,1}$ such that $(X,\left\| \cdot \right\| _{X,1})$ has both the Kadec-Klee and the Opial properties. In addition, $\left\| \cdot \right\| _{X,1}$ is uniformly convex in every direction and there exists a constant ${\tilde{K}}$ independent of X such that

$$\begin{aligned} \left\| \cdot \right\| _X \le \left\| \cdot \right\| _{X,1} \le {\tilde{K}}\;\left\| \cdot \right\| _X. \end{aligned}$$

It is known that no Banach space with the Opial property or which is uniformly convex in every direction or generally having normal structure can contain a diametrically complete set with empty interior ([42]), but in the case of an infinite-dimensional, reflexive and separable Banach space Theorem 5.1, when combined with the properties of Schauder bases and the Day norm, allows us to prove that each Banach space $(X, \Vert \cdot \Vert _X)$ of this kind has an equivalent norm $\Vert \cdot \Vert _0$ such that $(X, \Vert \cdot \Vert _0)$ is LUR and contains a diametrically complete set with empty interior [7].

In this section we enhance Theorem 5.1—we replace the Kadec-Klee property by local uniform rotundity. To this end, we first consider the case of the Banach space of real continuous functions on the interval [0, 1].

Theorem 5.2

Let $\epsilon$ be a positive real number. The Banach space $C([0,1],{\mathbb {R}})$ endowed with the standard max-norm $\left\| \cdot \right\| _C$ has an equivalent norm $\left\| \cdot \right\| _{C,I}$ such that $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _{C,I})$ has the Opial property, is LUR and uniformly convex in every direction, and $\left\| \cdot \right\| _C\le \left\| \cdot \right\| _{C,I}\le (2+\epsilon )\left\| \cdot \right\| _C.$

Proof

Let $\{{{\tilde{g}}}_i\}_{i=1}^\infty$ be the Schauder basis in $(C([0,1],{\mathbb {R}}), \left\| \cdot \right\| _C)$ ([47], see also [33]) and let $\{{{\tilde{g}}}^*_{_i}\}_{i=1}^\infty$ be the sequence of biorthogonal functionals associated with this basis. The basis $\{{{\tilde{g}}}_i\}_{i=1}^\infty$ is monotone, that is, if $\{P_n\}_{n=0}^\infty$ is the sequence of projections on $C([0,1],{\mathbb {R}})$, which are defined in the following way: $P_0:=0$, $P_n h := \sum _{i=1}^n {{\tilde{g}}}^*_{_i}(h) {{\tilde{g}}}_i$ for $n=1,2,...$ and $h \in C([0,1],{\mathbb {R}})$, then the sequence $\{\Vert P_n\Vert \}_{n=0}^\infty$ of the operator norms with respect to the norm $\left\| \cdot \right\| _{C}$ is bounded by 1. Hence the sequence $\{\Vert I-P_n\Vert \}_{n=0}^\infty$ of the operator norms is bounded by 2. Next, the sequence $\{\Vert {{\tilde{g}}}^*_{_i}\Vert \}_{i=1}^\infty$ is bounded with respect to the norm $\left\| \cdot \right\| _C$ by some $L>0$. Now, fix $\epsilon >0$ and choose $\eta >0$ such that

$$\begin{aligned} 0<\left( 1 + \frac{2}{3} \eta \right) \left[ 4+\frac{1}{3} \eta L^2\right] < (2 +\epsilon )^2. \end{aligned}$$

We introduce a new norm on $C([0,1],{\mathbb {R}})$ as follows:

$$\begin{aligned} \Vert h\Vert _{C,0}:=\sqrt{\Vert h\Vert _{{\mathcal {P}},C}^2 + \eta \Vert h\Vert _{2}^2}, \end{aligned}$$

where

$$\begin{aligned} \Vert h\Vert _{{\mathcal {P}},C} :=\sup _{n=0,1,2,...} \Vert (I -P_n) h\Vert _C= \sup _{n=0,1,2,...} \Vert h -P_n h\Vert _C \end{aligned}$$

and

$$\begin{aligned} \Vert h\Vert _2 :=\sqrt{\sum _{i=1}^\infty \left( \frac{g^*_{_i}(h)}{2^i}\right) ^2} \end{aligned}$$

for $h \in C([0,1],{\mathbb {R}})$. Then we have

• the norm $\left\| \cdot \right\| _{C,0}$ in $C([0,1],{\mathbb {R}})$ is equivalent to the norms $\left\| \cdot \right\| _C$ and $\left\| \cdot \right\| _{{\mathcal {P}},C},$ and

  $$\begin{aligned} \left\| \cdot \right\| _C \le \Vert \cdot \Vert _{{\mathcal {P}},C} \le \Vert \cdot \Vert _{C,0}= & {} \sqrt{\left[ \sup _{n=0,1,2,...} \Vert (I -P_n) (\cdot )\Vert _C\right] ^2 +\eta \sum _{i=1}^\infty \left( \frac{g^*_{_i}(\cdot )}{2^i}\right) ^2} \\{} & {} \quad \le \sqrt{4+\frac{1}{3} \eta L^2} \left\| \cdot \right\| _C. \end{aligned}$$

• the Banach space $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _{C,0})$ has the Opial property by Theorem 2.9 because the norm $\left\| \cdot \right\| _{{\mathcal {P}},C}$ has the nonstrict Opial property (see Theorem 2.10).

Next, let $\{X_i\}_{i=1}^\infty$ be an increasing sequence of finite dimensional subspaces of $C([0,1],{\mathbb {R}})$ such that $\bigcup _i X_i$ is dense in $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _C)$ and let $\{f_i^*\}_{i=1}^\infty$ be a separating (for $C([0,1],{\mathbb {R}})$) family of functionals with $\Vert f_i^*\Vert _{C^*,0} =1$ for each $i \in {\mathbb {N}}$ (the norm $\Vert \cdot \Vert _{C^*,0}$ in $C^*([0,1],{\mathbb {R}})$ is associated with the norm $\left\| \cdot \right\| _{C,0}$ in $C([0,1],{\mathbb {R}})$). Then the function given by

$$\begin{aligned} \left\| \cdot \right\| _{C,I} := \sqrt{\Vert \cdot \Vert _{C,0}^2 + \eta \sum _{i=1}^\infty \left( \frac{\textrm{dist}_{\Vert \cdot \Vert _{C,0}}(\cdot ,X_i)}{2^i}\right) ^2+\eta \sum _{i=1}^\infty \left( \frac{f_i^*(\cdot )}{2^i}\right) ^2} \end{aligned}$$

is a norm on $C([0,1],{\mathbb {R}})$ which is equivalent to the norm $\Vert \cdot \Vert _{C,0}$ and since $0 \in X_i$ for each $i \in {\mathbb {N}}$, we see that

$$\begin{aligned}{} & {} \left\| \cdot \right\| _C \le \Vert \cdot \Vert _{C,0} \le \left\| \cdot \right\| _{C,I} \\{} & {} \quad =\sqrt{\Vert \cdot \Vert _{C,0}^2 + \eta \sum _{i=1}^\infty \left( \frac{\textrm{dist}_{\Vert \cdot \Vert _{C,0}}(\cdot ,X_i)}{2^i}\right) ^2+\eta \sum _{i=1}^\infty \left( \frac{f_i^*(\cdot )}{2^i}\right) ^2} \\{} & {} \quad \le \sqrt{\left[ 4+\frac{1}{3} \eta L^2\right] \Vert \cdot \Vert _C^2 + \frac{1}{3} \eta \left[ 4+\frac{1}{3} \eta L^2\right] \Vert \cdot \Vert _C^2+\frac{1}{3} \eta \left[ 4+\frac{1}{3} \eta L^2\right] \Vert \cdot \Vert _C^2} \\{} & {} \quad = \sqrt{\left( 1 + \frac{2}{3}\eta \right) \left[ 4+\frac{1}{3} \eta L^2\right] }\Vert \cdot \Vert _C \le (2+\epsilon )\left\| \cdot \right\| _C. \end{aligned}$$

By Corollary 2.1 the norm $\left\| \cdot \right\| _{C,I}$ is uniformly convex in every direction. Next, by Theorem 2.6 the norm $\left\| \cdot \right\| _{C,I}$ is LUR. Hence we only need to show that this norm has the Opial property. So let $\{h_k\}_{k=1}^\infty$ be a weakly convergent null sequence. By separability of the Banach space $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _C)$ we may assume the existence of the limits

$$\begin{aligned}{} & {} \lim _k\Vert h_k - h\Vert _{C,I}, \\{} & {} \quad \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k-h,X_i) \end{aligned}$$

and

$$\begin{aligned} \lim _k \Vert h_k- h\Vert _{C,0} \end{aligned}$$

for each $h \in C([0,1],{\mathbb {R}})$. Observe here that by the Opial property of the norm $\Vert \cdot \Vert _{C,0}$ we have

$$\begin{aligned} \lim _k \Vert h_k\Vert _{C,0} \le \lim _k \Vert h_k- h\Vert _{C,0} \end{aligned}$$

for each $h \in C([0,1],{\mathbb {R}})$. Next, we show that

$$\begin{aligned} \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)= \lim _k \Vert h_k\Vert _{C,0}. \end{aligned}$$

Indeed, for each $i \in {\mathbb {N}}$ there exists a bounded sequence $\{\varphi _{k,i}\}_{k=1}^\infty$ in $X_i$ such that

$$\begin{aligned} \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)= \Vert h_k - \varphi _{k,i}\Vert _{C,0} \end{aligned}$$

for each $k \in {\mathbb {N}}$. Without loss of generality, we may assume that for each $i \in {\mathbb {N}}$,

$$\begin{aligned} \lim _k \varphi _{k,i} = \varphi _i \end{aligned}$$

in $((C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _{C,0})$, where $\varphi _i \in X_i$. Then we have

$$\begin{aligned} \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)= \lim _k \Vert h_k - \varphi _{k,i}\Vert _{C,0} = \lim _k \Vert h_k -\varphi _i\Vert _{C,0} \end{aligned}$$

for $i =1,2,...$. Now, since $0 \in X_i$ we obtain

$$\begin{aligned} \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i) \le \Vert h_k\Vert \end{aligned}$$

for $i,k \in {\mathbb {N}}$. Therefore the Opial property of the norm $\Vert \cdot \Vert _{C,0}$ (here we recall that $\{h_k\}_{k=1}^\infty$ is a weakly convergent null sequence) implies that

$$\begin{aligned} \lim _k \Vert h_k\Vert _{C,0} \le \lim _k \Vert h_k -\varphi _i\Vert _{C,0} = \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)\le \lim _k \Vert h_k \Vert _{C,0} \end{aligned}$$

for each $i \in {\mathbb {N}}$ and after applying the Opial property of the norm $\Vert \cdot \Vert _{C,0}$ once more we get $\varphi _i =0$ for $i=1,2....$ and

$$\begin{aligned} \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)= \lim _k \Vert h_k\Vert _{C,0} \end{aligned}$$

as we claimed. Next, take $0\ne {\tilde{h}} \in X$. We see that for each $i,k\in {\mathbb {N}}$, there exists $\psi _{k,i} \in X_i$ such that

$$\begin{aligned} \textrm{dist}_{\Vert _{C,0}}(h_k- {\tilde{h}},X_i)= \Vert h_k -{\tilde{h}} - \psi _{k,i}\Vert _{C,0} \end{aligned}$$

and we can choose a subsequence $\{k_l\}_{l=1}^\infty$ for which we have

$$\begin{aligned} \lim _l \Vert \psi _{k_l,i} - \psi _i\Vert _{C,0}=0, \end{aligned}$$

where $\psi _i \in X_i$ for each $i\in {\mathbb {N}}$. Then, by the Opial property of the norm $\Vert \cdot \Vert _{C,0}$, we obtain

$$\begin{aligned}{} & {} \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)=\lim _k \Vert h_k\Vert _{C,0} \\{} & {} \quad =\lim _l \Vert h_{k_l}\Vert _{C,0}\le \lim _l \Vert h_{k_l}- {\tilde{h}}- \psi _i\Vert _{C,0} = \lim _l \Vert h_{k_l}- {\tilde{h}}- \psi _{k_l,i}\Vert _{C,0} \\{} & {} \quad =\lim _l \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_{k_l}- {\tilde{h}},X_i)= \lim _k \textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k- {\tilde{h}},X_i) \end{aligned}$$

for each $i\in {\mathbb {N}}$ and consequently,

$$\begin{aligned} \lim _k \sum _{i=1}^\infty \left( \frac{\textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k,X_i)}{2^i}\right) ^2 \le \lim _k \sum _{i=1}^\infty \left( \frac{\textrm{dist}_{\Vert \cdot \Vert _{C,0}}(h_k- {\tilde{h}},X_i)}{2^i}\right) ^2. \end{aligned}$$

Finally, since $w-\lim _k h_k =0$, ${\tilde{h}}\ne 0$, and $\{f_i^*\}_{i=1}^\infty$ is a separating family for X, we get

$$\begin{aligned} \lim _k\sum _{i=1}^\infty \left( \frac{f_i^*(h_k)}{2^i}\right) ^2 =0 < \sum _{i=1}^\infty \left( \frac{f_i^*({\tilde{h}})}{2^i}\right) ^2 = \lim _k \sum _{i=1}^\infty \left( \frac{f_i^*(h_k -{\tilde{h}})}{2^i}\right) ^2. \end{aligned}$$

All the above inequalities imply the Opial property of the norm $\left\| \cdot \right\| _{X,I}$, as we claimed. $\square$

Remark 5.1

Theorem 5.2 can be proved differently. In this alternative proof we first obtain an equivalent norm with the Opial property (see Theorems 2.9 and 2.10) and next after the successive renorming (see Theorems 2.6) we get the claimed norm which has the Opial property, is LUR and uniformly convex in every direction. In this second proof we use Definition 2.3 of LUR instead of Definition 2.4 applied in the first proof. Finally, observe that these two different proofs of Theorem 5.2 are partially based on the ideas presented in [7, 13, 27, 52] and [54].

Now, as a corollary of Theorem 5.2 and the universality of $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _C)$ for the class of separable Banach spaces (see [22]), we can get the following general result. In the proof of this result we identify T(X) with X and write $X \subset C([0,1],{\mathbb {R}})$, when $(X,\left\| \cdot \right\| _X)$ is a Banach space, $C([0,1],{\mathbb {R}})$ is endowed with the standard max-norm $\left\| \cdot \right\| _C$ and $T:X \rightarrow C([0,1],{\mathbb {R}})$ is a linear isometric embedding of X into $C([0,1],{\mathbb {R}})$.

Theorem 5.3

Let $\epsilon$ be a positive real number. Every infinite-dimensional and separable Banach space $(X,\left\| \cdot \right\| _X)$ admits an equivalent norm $\left\| \cdot \right\| _{X,I}$ with both the LUR and the Opial properties, and such that

$$\begin{aligned} \left\| \cdot \right\| _X \le \left\| \cdot \right\| _{X,I} \le (2+\epsilon )\left\| \cdot \right\| _X. \end{aligned}$$

In addition, $\left\| \cdot \right\| _{X,I}$ is uniformly convex in every direction.

Proof

Fix $\epsilon >0$. By Theorem 5.2, the Banach space $C([0,1],{\mathbb {R}})$ endowed with the standard max-norm $\left\| \cdot \right\| _C$ has an equivalent norm $\left\| \cdot \right\| _{C,I}$ such that the space $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _{C,I})$ has both the LUR and the Opial properties, and

$$\begin{aligned} \left\| \cdot \right\| _C \le \left\| \cdot \right\| _{C,1} \le (2+\epsilon )\left\| \cdot \right\| _C. \end{aligned}$$

The space $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _{C,I})$ is also uniformly convex in every direction. By the universality of $(C([0,1],{\mathbb {R}}),\left\| \cdot \right\| _C)$ for the class of separable Banach spaces, we have $X \subset C([0,1],{\mathbb {R}})$. Now it is sufficient to take the restriction to X of the norm $\left\| \cdot \right\| _{C,I}$ in order to arrive at the asserted norm $\left\| \cdot \right\| _{X,I}$. $\square$

In view of Theorem 5.3 the following problem is quite natural.

Open problem Is 2 equal to the infimum of all $K >1$ satisfying the following condition:

for each separable Banach space $(X,\Vert \cdot \Vert _X)$ there exists an equivalent norm $\Vert \cdot \Vert _{X,\sim }$ such that

• $\Vert \cdot \Vert _X \le \Vert \cdot \Vert _{X,\sim } \le K \Vert \cdot \Vert _X,$

• $\Vert \cdot \Vert _{X,\sim }$ is LUR,

• $\Vert \cdot \Vert _{X,\sim }$ is uniformly convex in every direction,

• $\Vert \cdot \Vert _{X,\sim }$ has the Opial property?