Locally Nearly Uniformly Convex Points in Orlicz Spaces Equipped with the Luxemburg Norm

Abstract

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. Then, we analyze the intrinsic connection between locally nearly uniformly convex points and nearly extreme points in Banach spaces. Additionally, we provide comprehensive characterizations of locally nearly uniformly convex points in both Orlicz function spaces and Orlicz sequence spaces under the Luxemburg norm. These findings enrich the geometric theory system of Banach and Orlicz spaces, offering new theoretical support for related research directions.

1. Introduction

Let $X$ be a Banach space with the open unit ball $U(X)$, the closed unit ball $B(X)$ and the unit sphere $S(X)$, respectively, and let $X^{\ast }$ be the dual space of $X$.

By $U(X^{\ast })$, $B(X^{\ast })$ and $S(X^{\ast })$, we note the open unit ball, the closed unit ball and the unit sphere of $X^{\ast }$, respectively. By $w$, $w^{\ast }$ and $\Vert \cdot \Vert$, we note topologies $w$ and $w^{\ast }$, and norm, respectively. By $co(A)$ and $\overline{co}(A)$, we note the convex hull and the closed convex hull of the set $A$, respectively. By $x_n\stackrel{w}{⟶}x$ we denote that a sequence $\left\{x_n\right\}\subset X$ is weakly converging to $x\in X$. We proceed to define the following:

$$sep(x_n)=\inf \{\parallel x_n-x_m\parallel :m\ne n\}.$$

$$B_{\delta }(0)=\{x:\Vert x\Vert \le \delta \}.$$

for $f\in X^{\ast }$, we define

$$A_f=\{x\in S(X):f(x)=1=\parallel f\parallel \}.$$

For $x\in X$, we define

$$D_x=\{f\in S(X^{\ast }):f(x)=1=\parallel x\parallel \}.$$

The geometric properties of Banach spaces have long been a core research topic in functional analysis, with convexity-related properties serving as key tools in fields such as fixed point theory, approximation theory, and optimization. Since M.M. Day proposed the $(KK)$ property in 1973 [1], scholars worldwide have conducted in-depth explorations of generalized convexity in Banach spaces.

A Banach space $X$ is said to have the Kadec–Klee $(KK)$ property for any $x\in S(X)$, $\left\{x_n\right\}\subset X$, if $x_n\stackrel{w}{\to }x$, then $\parallel x_n-x\parallel \to 0$.

In 1980, Huff [2] introduced the concepts of $(UKK)$ and $(NUC)$ spaces.

We say that a Banach space $X$ has the $(UKK)$ property if for every $\epsilon >0$, there exists $0<\delta <1$ such that for any sequence $\left\{x_n\right\}\subset U(X)$ with $sep(x_n)\ge \epsilon$, if $x_n\stackrel{w}{\to }x$ and $sep(x_n)\ge \epsilon$, then $x\in B_{\delta }(0)$.

A Banach space $X$ is called a $(NUC)$ space if for every $\epsilon >0$, there exists $0<\delta =\delta (\epsilon )<1$ such that for any sequence $\left\{x_n\right\}\subset U(X)$ with $sep(x_n)\ge \epsilon$, then there is a $N\ge 1$ and scalars ${\lambda }_1,\cdots ,{\lambda }_N\ge 0$ with ${\displaystyle \sum _{n=1}^N{\lambda }_n}=1$ such that $\parallel \sum {\lambda }_n{x}_n\parallel \le 1-\delta$.

Huff also showed that a Banach space $X$ is $(NUC)$ if and only if $X$ is $(UKK)$ and reflexive.

It is widely acknowledged that the condition equivalent to $(NUC)$ was put forward independently in [3,4].

In 1948, Brodskiǐ and Milman [5] introduced the concept of normal structure and weak normal structure.

A bounded, convex subset $K$ of Banach space $X$ is said to have normal structure if every convex subset $H$ of $K$ that contains more than one point contains a point $x_0\in H$, such that

$$\sup \{\parallel x_0-y\parallel :y\in H\}<d(H),$$

where

$$d(H)=\sup \left\{\parallel x-y\parallel :x,y\in H\right\},$$

denotes the diameter of $H$.

A Banach space $X$ is said to have normal structure if every bounded, convex subset of $X$ has normal structure.

A Banach space $X$ is said to have weak normal structure if for each weakly compact convex set $K$ in $X$ that contains more than one point has normal structure. W.A. Kirk’s research on non-expansive mappings [6] laid an important foundation for the application of convexity properties, while Sekowski and Stachura introduced the concept of nearly strict convex Banach spaces in 1988 [7], expanding the research scope of convexity.

In a Banach space $X$, $x\in S(X)$ is a nearly extreme point of $B(X)$, if there does not exists any non-compact convex subset in $S(X)$ which contains $x$.

A Banach space $X$ is nearly strict convex if every $x\in S(X)$ is a nearly extreme point.

Nan Chaoxun and Song Shoubai obtained the following significant results on the compactness of specific sets in nearly strict convex spaces in 1997 [8], further promoting the development of this field.

A $x\in S(X)$ is a nearly extreme point if and only if for any $f\in D_x$, the set $A_f$ is compact.

In 1990, D.N. Kutzarova put forward new convexity-related concepts [9], and Rolewicz made important contributions to the study of A-uniform convexity and the drop property [10]. Rolewicz showed that the property $(\beta )$ follows from the uniform convexity, and that the property $(\beta )$ implies $(NUC)$. Moreover, the property $(\beta )$ is isomorphically different from both of them (see [11,12,13]).

For any subset $C$ of $X$, the Kuratowski measure $\alpha$ of $C$ $(\alpha (C))$ is the infimum of those $\epsilon >0$ for which there is a covering of $C$ by a finite number of sets of diameter less than $\epsilon$.

For any $x\notin B(X)$, the drop determined by $x$ is the set

$$D(x,B(X))=co(\{x\}\bigcup B(X)).$$

We say that a Banach space $X$ has property $(\beta )$ if for any $\epsilon >0$, there exists $\delta >0$ such that $\alpha (D(x,B(X))\backslash B(X))<\epsilon$ whenever $1<\parallel x\parallel <1+\delta$.

D.N. Kutzarova introduced the following equivalent form of the property $(\beta )$:

A Banach space $X$ has the property $(\beta )$ if and only if for every $\epsilon >0$, there exists $\delta >0$ such that for each element $x\in B(X)$ and each sequence $\left\{x_n\right\}\subset B(X)$ with $sep(x_n)\ge \epsilon$ there is an index $k$ for which $‖{\displaystyle \frac{x+x_k}{2}}‖\le 1-\delta$.

The exploration of property $(\beta )$ and its local form has also attracted extensive attention: Cui Yunan, Ryszard Pluciennik, and Wang Tingfu studied the equivalent form of property $(\beta )$ in Orlicz spaces in 1997 [14]; Wang Tingfu, Cui Yunan, and Meng Chenghui defined local $(\beta )$ points in 2000 [15]; and Shang Shaogiang and Cui Yunan introduced the concept of almost convexity and conducted in-depth research in Banach spaces in 2021 [16].

A $x\in S(X)$ is said to be a local $(\beta )$ point if for any $\epsilon >0$, there exists $\delta =\delta (x,\epsilon )\in (0,1)$ such that for any sequence $\left\{x_n\right\}\subset S(X)$ with $sep(x_n)\ge \epsilon$, there holds a $n_0\in \mathbb{N}$ for which $||{\displaystyle \frac{x_{n_0}+x}{2}}||<1-\delta$.

The concept of Local Nearly Uniformly Convex $(LNUC)$ spaces, proposed by D.N. Kutzarova and B.L. Lin in 1994 [17], is an important generalization of nearly uniform convexity.

A Banach space $X$ is called a Local Nearly Uniformly Convex $(LNUC)$ space if for every $\epsilon >0$ and $x\in U(X)$ there exists $\delta =\delta (x,\epsilon )>0$ such that for any sequence $\left\{x_n\right\}\subset U(X)$ with $sep(x_n)\ge \epsilon$, then,

$$co(\{x_n,x\})\bigcap B_{\delta }(0)\ne \varphi .$$

In other words,

$$\begin{array}{c}\parallel {\lambda }_{0}y+(1-{\lambda }_0)x\parallel <\delta \end{array},$$

for some ${\lambda }_0\in (0,1)$ and $\begin{array}{c}y\in co(\{x_n\})\end{array}$.

Orlicz spaces, as a typical class of Banach spaces, exhibit unique geometric structures, and the study of their convexity properties is of great theoretical significance. This paper focuses on nearly convex points and locally nearly uniformly convex points, aiming to establish their characterization conditions in Orlicz spaces with the Luxemburg norm, thereby advancing the development of the geometric theory of Orlicz spaces.

Definition 1.

In a Banach space $X$, a $x\in S(X)$ is a nearly convex point, if for $\left\{x_n\right\}\subset S(X)$ with $sep(x_n)>0$, then there exist $\lambda \in (0,1)$ and $x_{n_0}\in \left\{x_n\right\}$ satisfying

$$\begin{array}{c}\parallel \lambda x+(1-\lambda )x_{n_0}\parallel <1\end{array}.$$

(1)

Definition 2.

In a Banach space  $X$, a $x\in S(X)$ is a locally nearly uniformly convex point, if for any $\epsilon >0$ there exists $0<\delta <1$ such that for any $\left\{x_n\right\}\subset S(X)$ with $sep(x_n)>\epsilon$, there hold $\lambda \in (0,1)$ and $n_0\in \mathbb{N}$ for which

$$\begin{array}{c}\parallel \lambda x_{n_0}+(1-\lambda )x\parallel <1\end{array}-\delta .$$

(2)

It is clear that if $x\in S(X)$ is a local $(\beta )$ point, then $x$ is a locally nearly uniformly convex point.

One aim of this study is to give a criterion that an Orlicz space equipped with the Luxemburg norm is locally uniformly nearly convex. Some basic facts about Orlicz spaces follow.

A mapping $\mathsf{\Phi }$ is called an Orlicz function provided that $\mathsf{\Phi }:ℝ\to {ℝ}^{+}=[0,+\infty ]$ is even, convex, left-continuous on $[0,+\infty )$ with $\mathsf{\Phi }(0)=0$, and there exists $u_0>0$ such that $\mathsf{\Phi }(u_0)<+\infty$. We define: $a(\mathsf{\Phi })=\sup \left\{u\ge 0:\mathsf{\Phi }(u)=0\right\}$ and $b(\mathsf{\Phi })=\sup \left\{u\ge 0:\mathsf{\Phi }(u)<+\infty \right\}$ (see [11,12,13]).

By $\mathsf{\Psi }$ we note the complement to $\mathsf{\Phi }$ in the sense of Young, i.e.,

$$\mathsf{\Psi }(v)=\sup \left\{u\left|v\right|-\mathsf{\Phi }(u):u\ge 0\right\},$$

for each $v\in ℝ$.

Let $(G,\Sigma ,m)$ be a measure space with a finite measure $m$. Denote by $L^0(G)$ the set of all $m-$ equivalence classes of real valued measurable functions defined on $G$. The Orlicz function space $L_{\mathsf{\Phi }}(G)$ is defined as

$$L_{\mathsf{\Phi }}(G)=\{x\in L^0(G):I_{\mathsf{\Phi }}(\lambda x)={{\displaystyle \int }}_G\mathsf{\Phi }\left(\lambda x(t)\right)dm<+\infty ,\mathrm{for} \mathrm{some} \lambda >0\}.$$

The Luxemburg norm on $L_{\mathsf{\Phi }}(G)$ is defined as

$${\parallel x\parallel }_{\mathsf{\Phi }}=\inf \left\{\lambda >0:I_{\mathsf{\Phi }}\left({\displaystyle \frac{x}{\lambda }}\right)\le 1.\right\}$$

(3)

Denote by $l^0$ the set of all real sequences. Orlicz sequence space $l_{\mathsf{\Phi }}$ is defined as $l_{\mathsf{\Phi }}=\{x\in l^0:I_{\mathsf{\Phi }}(\lambda x)={\displaystyle \sum _{i=1}^{\infty }}\mathsf{\Phi }(\lambda x(i))<+\infty ,$ for some $\lambda >0\}$.

The Luxemburg norm on $l_{\mathsf{\Phi }}$ is defined as

$${\parallel x\parallel }_{\mathsf{\Phi }}=\inf \left\{\lambda >0:I_{\mathsf{\Phi }}\left({\displaystyle \frac{x}{\lambda }}\right)\le 1\right\}.$$

(4)

$p(u)$ and $p_{-}(u)$ stand for the right and left derivative of $\mathsf{\Phi }$ at $u\in ℝ$. An interval $[a,b]$ is called a structurally affine interval of $\mathsf{\Phi }$. Provided that $\mathsf{\Phi }$ is affine on $[a,b]$ and it is not affine either on $[a-\epsilon ,b]$ or on $[a,b+\epsilon ]$ for any $\epsilon >0$.

Put

$$SAI(\mathsf{\Phi })=\{[a_1,b_1],[a_2,b_2],\cdots ,[a_m,b_m]\},$$

where $[a_i,b_i](i=1,\cdots ,m)$ are structurally intervals.

$$S{C}_{\mathsf{\Phi }}^{-}=\{u\in ℝ:\mathrm{there} \mathrm{exists} \epsilon >0\mathrm{such} \mathrm{that} \mathsf{\Phi }\mathrm{is} \mathrm{a} \mathrm{affine} \mathrm{on} [u-\epsilon ,u]\}.$$

An Orlicz function $\mathsf{\Phi }$ satisfies the ${\Delta }_2$-condition (denoted as $\mathsf{\Phi }\in {\Delta }_2$) if there exist constants $k\ge 2$ and $u_0\ge 0$ such that $\mathsf{\Phi }(2u)\le k\mathsf{\Phi }(u)$ for all $u\ge u_0$.

We say $\mathsf{\Phi }\in {\nabla }_2$ if its complementary function $\mathsf{\Psi }$ satisfies the ${\Delta }_2$-condition.

An Orlicz function $\mathsf{\Phi }$ is said to satisfy the ${\delta }_2$-condition ($\mathsf{\Phi }\in {\delta }_2$ for short) if there exist constants $k\ge 2$ and $u_0\ge 0$ such that $\mathsf{\Phi }(2u)\le k\mathsf{\Phi }(u)$ for every $\left|u\right|\le u_0$.

An Orlicz function $\mathsf{\Phi }$ is said to satisfy the ${\delta }_2^0$-condition ($\mathsf{\Phi }\in {\delta }_2^0$ for short) if its complementary function $\mathsf{\Psi }$ satisfies the ${\delta }_2$-condition. $\mathsf{\Phi }\in {\delta }_2^0$ if and only if $\mathsf{\Psi }\in {\delta }_2$.

For details on Orlicz space, please see [18].

2. Materials and Methods

In this study, we introduce the nearly convex point and the locally nearly uniformly convex point. Then we provide the general characterization of a nearly convex point in Banach space. The relationship between a locally nearly uniformly convex point and the nearly extreme point is also investigated in Banach space. Finally, we give the criteria that Orlicz function spaces and Orlicz sequence spaces equipped with Luxemburg norm are locally nearly uniformly convex, respectively.

3. Main Results

3.1. Preliminaries

Let us recall the following lemma, which will be used in the remainder of the paper.

For $x\in L_{\mathsf{\Phi }}(G)$ and $n\in \mathbb{N}$, we define

$$x_n(t)=\left\{\begin{array}{cc}x(t),\hfill & \hfill |x(t)|⩽n\\ 0,\hfill & \hfill |x(t)|>n\end{array}\right.$$

(5)

$$Q_{\mathsf{\Phi }}(x)=\inf \{c>0:I_{\mathsf{\Phi }}\left({\displaystyle \frac{x}{c}}\right)<\infty \}=\underset{n\to \infty }{\lim }\parallel x-x_n{\parallel }_{\mathsf{\Phi }}$$

(6)

(see [18]).

By $[a,b]$ we denote the affine interval of $\mathsf{\Phi }$ and by $(a_i,b_i)$ we denote the set of all affine intervals of $\mathsf{\Phi }(u)$.

Lemma 1 ([17]).

In a Banach space $X$, $x\in S(X)$ is a nearly extreme point if and only if

(i) 

$Q_{\mathsf{\Phi }}(x)<1$;

(ii) 

The measure  $m(\{x\in G:x(t)\in {\displaystyle \bigcup _{i=1}^m(a_i,b_i)\})}=0$;

(iii) 

The measure  $m(\{x\in G:x(t)\in {\displaystyle \bigcup _{i=1}^m\left\{a_i\right\}\})}\cdot m(\{x\in G:x(t)\in {\displaystyle \bigcup _{i=1}^m\left\{b_i\right\}\})}=0$.

3.2. Characterizations in Banach Spaces

We present the main definitions and theorems of this section.

Theorem 1.

Let  $X$ be a Banach space, if $x\in S(X)$ is a nearly convex point, then for any $\left\{x_n\right\}\subset S(X)$ with $sep(x_n)>0$, there exists $\begin{array}{c}{n}_0\in N\end{array}$ satisfying

$$co\begin{array}{c}(x,x_{n_0})=\{\lambda x+(1-\lambda )x_{n_0}:0<\lambda <1\end{array}\}\subset U(X).$$

(7)

Proof of Theorem 1.

Since $x$ is a locally nearly convex point, there exist $\begin{array}{c}{n}_0\in \mathbb{N}\end{array}$ and ${\lambda }_0\in (0,1)$ such that

$$‖{\lambda }_{0}x+(1-{\lambda }_0)x_{n_0}‖<1.$$

(8)

We need to verify that

$$‖\lambda x+(1-\lambda )x_{n_0}‖<1,$$

(9)

for any $\lambda \in (0,1)$. Assume for contradiction that there exists ${\lambda }_1\in (0,1)$ satisfying

$$‖{\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}‖=1.$$

(10)

Then

$${\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}\in (x,{\lambda }_{0}x+(1-{\lambda }_0)x_{n_0}),$$

(11)

or

$${\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}\in ({\lambda }_{0}x+(1-{\lambda }_0)x_{n_0},x).$$

(12)

We may assume that ${\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}\in (x,{\lambda }_{0}x+(1-{\lambda }_0)x_{n_0})$, so there exists ${\lambda }_2\in (0,1)$ such that

$${\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}={\lambda }_{2}x+(1-{\lambda }_2)({\lambda }_{0}x+(1-{\lambda }_0)x_{n_0}).$$

(13)

By the convexity of the norm, we have

$$\begin{array}{l}\Vert {\lambda }_{1}x+(1-{\lambda }_1)x_{n_0}\Vert \\ \le {\lambda }_2\Vert x\Vert +(1-{\lambda }_2)\Vert {\lambda }_{0}x+(1-{\lambda }_0)x_{n_0}\Vert \\ <{\lambda }_2+(1-{\lambda }_2)=1,\end{array}$$

(14)

This contradicts $\Vert {\lambda }_{0}x+(1-{\lambda }_0)x_{n_0}\Vert =1$, so the conclusion holds. □

Theorem 2.

Let  $X$ be a Banach space, if $x\in S(X)$ is a locally nearly uniformly convex point, then $x$ is a nearly extreme point.

Proof of Theorem 2.

Suppose that $x\in S(X)$ is not a nearly extreme point, then there exists a $f\begin{array}{c}\in D_x\end{array}$ for which $A_f$ is non-compact. Hence, there exists a sequence $\left\{x_n\right\}\subset A_f$ with $sep(x_n)>0$. Therefore, we have

$$\begin{array}{l}f(\lambda x_n+(1-\lambda )x)\\ =\lambda f(x_n)+(1-\lambda )f(x)\\ =\lambda \Vert x\Vert +(1-\lambda )\Vert x\Vert =1,\end{array}$$

(15)

for any $\lambda \in (0,1)$, i.e.,

$$‖\lambda x_n+(1-\lambda )x‖=1,$$

(16)

for any $n\in \mathbb{N}$ and $\lambda \in (0,1)$. However, since $x$ is a locally nearly convex point, there exist $\lambda \in (0,1)$ and $n_0\in \mathbb{N}$ such that $\Vert \lambda x_n+(1-\lambda )x\Vert <1,$ which is a contradiction, and, therefore, $x$ is a nearly extreme point. □

3.3. Characterizations in Orlicz Function Spaces

Theorem 3.

A point  $x\in S(L_{\mathsf{\Phi }}(G))$ is a locally nearly convex point if and only if the following conditions are satisfied:

(1) 

$\mathsf{\Phi }\in {\Delta }_2$;

(2) 

$b(\mathsf{\Phi })=+\infty$;

(3) 

$m(\{t\in G:0<\left|x(t)\right|\le a(\mathsf{\Phi })\})=0$

(4) 

$m(\{t\in G:x(t)\in SAI(\mathsf{\Phi })\})=0$;

(5) 

$m(\{t\in G:x(t)\in {\left\{a_n\right\}}_{n=1}^m\})=0$ and $\mathsf{\Phi }\in {\nabla }_2$ or $m(\{t\in G:x(t)\in {\left\{b_n\right\}}_{n=1}^m\})=0$ where $\left\{a_n\right\}$ and $\left\{b_n\right\}$ are the endpoints of the affine intervals of $\mathsf{\Phi }$.

Proof of Theorem 3.

Since $(1)+(2)+(3)+(4)+(5)$, there hold that $x$ is a local $(\beta )$ point. Then $x$ is a locally nearly uniformly convex point (see [15]). Hence, we only need to prove the necessity.

(1) Take $a>0$ such that $m(\{t\in G:\left|x(t)\right|\le a\})>0$, put $G_a=\{t\in G:|x(t)|\le a\}$. Then $x{\chi }_{G_a}\in E_{\mathsf{\Phi }}(G)$. Suppose $\mathsf{\Phi }\notin {\Delta }_2$, then there exists a monotonically increasing sequence $\left\{u_n\right\}$ with $\underset{n\to \infty }{\lim }{u}_n=+\infty$ such that

$$\mathsf{\Phi }\left((1+{\displaystyle \frac{1}{n}})u_n\right)\ge 2^n\mathsf{\Phi }(u_n) \mathrm{for} n=1,2,\cdots$$

(17)

Take a disjoint subset $\left\{G_n\right\}\subset G_a$ such that

$$\mathsf{\Phi }(u_n)m(G_n)={\displaystyle \frac{1}{2^n}} \mathrm{for} n=1,2,\cdots$$

(18)

Take a subset $E\subset G_a$ such that $0<{{\displaystyle \int }}_E\mathsf{\Phi }\left(x\left(t\right)\right)dt\le {\displaystyle \frac{1}{2}}$.

Define

$$x_n\left(t\right)=x(t){\chi }_E+\sum _{i=n}^{\infty }{u}_i{\chi }_{G_i}\mathrm{for} n\in \mathbb{N}.$$

(19)

Then

$$\begin{array}{cc}\hfill I_{\mathsf{\Phi }}\left(x_n\right)& ={\displaystyle {\int }_E\mathsf{\Phi }\left(x(t)\right)}dt+{\displaystyle \sum _{i=n}^{\infty }}\mathsf{\Phi }\left(u_i\right)m\left(G_i\right)\hfill \\ & \le 1-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}=1,\hfill \end{array}$$

(20)

and for any $\epsilon \in (0,1)$, there exists $i_0\in \mathbb{N}$ such that

$$1+{\displaystyle \frac{1}{i_0}}\le {\displaystyle \frac{1}{\epsilon }}.$$

(21)

Hence,

$$\begin{array}{l}{I}_{\mathsf{\Phi }}\left({\displaystyle \frac{x_n}{\epsilon }}\right)\ge {\displaystyle \sum _{i=\max \{i_0,n\}}}\mathsf{\Phi }\left((1+{\displaystyle \frac{1}{i}})u_i\right)m(G_i)\\ ={\displaystyle \sum _{i=\max \{i_0,n\}}^n}{2}^i\mathsf{\Phi }(u_i)m(G_i)\\ =+\infty .\end{array}$$

(22)

(2) It is easy to verify that if $\mathsf{\Phi }\in {\Delta }_2$, then $b(\mathsf{\Phi })=+\infty$. Using $\mathsf{\Phi }\in {\Delta }_2$, we can get $b(\mathsf{\Phi })=+\infty$. Otherwise, if $b(\mathsf{\Phi })<+\infty$, take

$$u_n=\left(1-{\displaystyle \frac{1}{n}}\right)b(\mathsf{\Phi }),$$

(23)

then $2u_n>b(\mathsf{\Phi })$. For any monotonically increasing sequence $\left\{k_n\right\}$ with $\underset{n\to +\infty }{\lim }{k}_n=+\infty$, we have

$$k_n\mathsf{\Phi }(u_n)<\mathsf{\Phi }(2u_n)=+\infty ,$$

(24)

a contradiction.

(3) Suppose the conditions are false.

Put

$$G_0=\{t\in G:0<|x(t)|\le a(\mathsf{\Phi })\}.$$

(25)

Then

$$m(G_0)>0.$$

(26)

Take an interval $[c,d]\subset (0,a(\mathsf{\Phi }))$ and two disjoint subsets $G_{1,1},G_{1,2}$ with

$$m(G_{1,1})=m(G_{1,2})={\displaystyle \frac{1}{2}}m(G_0).$$

(27)

Define

$$x_1(t)=c{\chi }_{G_{1,1}}+d{\chi }_{G_{1,2}}+x(t){\chi }_{G|G_0}.$$

(28)

Divide $G_{1,1}$ into two disjoint subsets $G_{2,1},G_{2,2}$, divide $G_{1,2}$ into two disjoint subsets $G_{2,3},G_{2,4}$ with

$$m(G_{2,1})=m(G_{2,2})=m(G_{2,3})=m(G_{2,4})={\displaystyle \frac{1}{2^2}}m(G_0)$$

(29)

Define

$$\begin{array}{l}{x}_2(t)=c{\chi }_{G_{2,1}}+d{\chi }_{G_{2,2}}+c{\chi }_{G_{2,3}}+d{\chi }_{G_{2,4}}+x(t)\\ =c{\chi }_{{\displaystyle \bigcup _{i=1}^2{G}_{2,2i-1}}}+d{\chi }_{{\displaystyle \bigcup _{i=1}^2{G}_{2i}}}+x(t){\chi }_{G|G_0}.\end{array}$$

(30)

In such a way, we get a sequence of subsets ${\left\{G_{n,i}\right\}}_{i=1}^{2n}$ such that

(i)

$G_{n,i}\bigcap G_{n,j}\ne \varphi ,i\ne j$;

(ii)

$m(G_{n,i})={\displaystyle \frac{1}{2^n}}m(G_0)$.

• Define

$$x_n(t)=c{\chi }_{{\displaystyle \bigcup _{i=1}^{2^n}{G}_{n,2i-1}}}+d{\chi }_{{\displaystyle \bigcup _{i=1}^{2^n}{G}_{n,2i}}}+x(t){\chi }_{G|G_0},$$

(31)

then $I_{\mathsf{\Phi }}(x_n)=I_{\mathsf{\Phi }}(x)=1$,

$${\Vert x_n\Vert }_{\mathsf{\Phi }}=1.$$

(32)

for each $n\in \mathbb{N}$. It is clear that

$${\Vert x_n-x_m\Vert }_{\mathsf{\Phi }}={\displaystyle \frac{d-c}{{\mathsf{\Phi }}^{-1}\left(\frac{2}{m(G_0)}\right)}}.$$

(33)

for any $\lambda \in (0,1)$

$$I_{\mathsf{\Phi }}(\lambda x_n+(1-\lambda )x(t))=I_{\mathsf{\Phi }}(x)=1$$

(34)

i.e.,

$${\Vert \lambda x_n-(1-\lambda )x\Vert }_{\mathsf{\Phi }}=1,$$

(35)

contradiction.

(4) By Lemma 1, we can obtain

$$m(\{t\in G:x(t)\in {\displaystyle \bigcup _{i=1}^m\left\{a_i\right\}\})}\cdot m(\{t\in G:x(t)\in {\displaystyle \bigcup _{i=1}^m\left\{b_i\right\}\})}=0.$$

(36)

(5) Suppose $\mathsf{\Phi }\notin {\nabla }_2$ and $m(\{t\in G:|x(t)|\in {\displaystyle \bigcup _{i=1}^m\left\{b_i\right\}\})}>0$. Using $\mathsf{\Phi }\notin {\nabla }_2$, then for any $\lambda \in (0,1)$ There exists a monotonically increasing sequence $\left\{v_n\right\}$ such that

$$\mathsf{\Phi }(\lambda v_n)\ge \lambda \left(1-{\displaystyle \frac{1}{n}}\right)\mathsf{\Phi }(v_n),$$

(37)

for each $n\in \mathbb{N}$ there is a $n_0\in [1,m]$ such that $m(\{t\in G:x(t)=b_{n_0}\})>0$ (we may assume that $x(t)\ge 0$).

$$E_0=\{t\in G:x(t)=b_{n_0}\}.$$

(38)

Put $\epsilon >0$ such that $b_{n_0}-\epsilon >a_{n_0}$, $G_n\subset E_0$ with $G_n\cap G_m=\varphi (n\ne m)$,

$$\mathsf{\Phi }\left(v_n-{\displaystyle \frac{1-\lambda }{\lambda }}{b}_{n_0}\right)m(G_n)=\lambda \epsilon P(b_{n_0})m(E_0 )(n\in \mathbb{N}).$$

(39)

By $\underset{n\to \infty }{\lim }\mathsf{\Phi }\left(v_n-b_{n_0}\right)=+\infty$, we get $\underset{n\to +\infty }{\lim }m\left(G_n\right)=0$.

For convenience, we write $b_{n_0}=b$. Define

$$x_n(t)=x(t){\chi }_{G|E_0}+(b-\epsilon ){\chi }_{E_0|G_n}+\left(v_n-{\displaystyle \frac{1-\lambda }{\lambda }}b\right){\chi }_{G_n}.$$

(40)

Then

$$\begin{array}{cc}\hfill I_{\mathsf{\Phi }}(x_n)& =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\mathsf{\Phi }(1-\epsilon )m(E_0\left|G_n\right.)+\mathsf{\Phi }({\nu }_n-b)m(G_n)\hfill \\ & =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+({\int }_0^{b-\epsilon }{p}_{-}(t)dt)m(E_0\left|G_n\right.)+\mathsf{\Phi }({\nu }_n-b)m(G_n)\hfill \\ & =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+({\int }_0^b{p}_{-}(t)dt+{\int }_{b-\epsilon }^b{p}_{-}(t)dt)m(E_0\left|G_n\right.)+\epsilon {\mathrm{p}}_{-}(b)m(E_0)\hfill \\ & =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+(\mathsf{\Phi }(b)-\epsilon {\mathrm{p}}_{-}(b))m(E_0\left|G_n\right.)+\epsilon {\mathrm{p}}_{-}(b)m(E_0)\hfill \\ & =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+I_{\mathsf{\Phi }}(x{\chi }_{E_0})-\mathsf{\Phi }(b)m(G_n)+\epsilon {\mathrm{p}}_{-}(b)m(G_n)\hfill \\ & =1-\mathsf{\Phi }(b)m(G_n)+\epsilon {\mathrm{p}}_{-}(b)m(G_n)\hfill \\ & \to 1 as n\to \infty .\hfill \end{array}$$

(41)

By $\mathsf{\Phi }\in {\Delta }_2$ we get that ${\left|\left|x_n\right|\right|}_{\mathsf{\Phi }}\to 1$ as $n\to +\infty$.

In virtue of that

$$\begin{array}{l}{I}_{\mathsf{\Phi }}(\lambda x_n+(1-\lambda )x)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\mathsf{\Phi }(\lambda (b-\epsilon )+(1-\lambda )b)m(E_0\left|G_n\right.)\\ +\mathsf{\Phi }\left(\lambda (v_n-{\displaystyle \frac{1-\lambda }{\lambda }}b)+(1-\lambda )b\right)m(G_n)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\mathsf{\Phi }(b-\lambda \epsilon )m(E_0\left|G_n\right.)+\mathsf{\Phi }(\lambda v_n)m(G_n)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+{\int }_0^{b-\lambda \epsilon }{p}_{-}(t)dtm(E_0\left|G_n\right.)+\mathsf{\Phi }(\lambda {\nu }_n)m(G_n)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+({\int }_0^b{p}_{-}(t)dt+{\int }_{b-\lambda \epsilon }^b{p}_{-}(t))m(E_0\left|G_n\right.)+\mathsf{\Phi }(\lambda {\nu }_n)m(G_n)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+(\mathsf{\Phi }(b)-\lambda \epsilon {\mathrm{p}}_{-}(b))m(E_0\left|G_n\right.)+\mathsf{\Phi }(\lambda {\nu }_n)m(G_n)\\ \ge I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+(\mathsf{\Phi }(b)-\lambda \epsilon {\mathrm{p}}_{-}(b))m(E_0\left|G_n\right.)+(1-{\displaystyle \frac{1}{n}})\mathsf{\Phi }({\nu }_n)m(G_n)\\ \ge I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\mathsf{\Phi }(b)m(E_0\left|G_n\right.)-\lambda \epsilon {\mathrm{p}}_{-}(b)m(E_0)+\lambda \epsilon {\mathrm{p}}_{-}(b)m(G_n)\\ +(1-{\displaystyle \frac{1}{n}})\mathsf{\Phi }\left({\nu }_n-{\displaystyle \frac{1-\lambda }{\lambda }}b\right)m(G_n)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\underset{E_0|G_a}{\int }\mathsf{\Phi }\left(x(t)\right)dt-\lambda \epsilon {\mathrm{p}}_{-}(b)m(E_0)+\lambda \epsilon {\mathrm{p}}_{-}(b)m(G_n)\\ +(1-{\displaystyle \frac{1}{n}})\lambda \epsilon {\mathrm{p}}_{-}(b)m(E_0)\\ =I_{\mathsf{\Phi }}(x{\chi }_{G|E_0})+\lambda \epsilon {\mathrm{p}}_{-}(b)m(G_n)-{\displaystyle \frac{1}{n}}\lambda \epsilon {\mathrm{p}}_{-}(b)m(E_0)\to 1 as n \to +\infty ,\end{array}$$

(42)

thanks to $\underset{n\to +\infty }{\lim }m\left(G_n\right)=0$ and $\mathsf{\Phi }\in {\Delta }_2$, we get that

$${\Vert \lambda x_n-(1-\lambda )x\Vert }_{\mathsf{\Phi }}\to 1 as n\to +\infty .$$

(43)

By $\mathsf{\Phi }\in {\Delta }_2$, there exists a $\delta >0$ such that

$$|I_{\mathsf{\Phi }}(u)+I_{\mathsf{\Phi }}(u-v)|<{\displaystyle \frac{\epsilon }{2}}P(b)m(G_0),$$

(44)

whenever $I_{\mathsf{\Phi }}(u)\le 1$ and $I_{\mathsf{\Phi }}(v)\le \delta$.

Since $\underset{n\to +\infty }{\lim }\mathsf{\Phi }(b-\lambda \epsilon )m\left(G_n\right)=0$, there is a $n_0\in \mathbb{N}$ such that

$$I_{\mathsf{\Phi }}(b-\lambda \epsilon ){\chi }_{G_n}=\mathsf{\Phi }(b-\lambda \epsilon )m(G_n)<\delta ,$$

(45)

Whenever $n\ge n_0$.

Thus, for any $n>m\ge n_0$, we have

$$\begin{array}{l}{I}_{\mathsf{\Phi }}(x_n-x_m)\\ \ge \mathsf{\Phi }\left(v_n-{\displaystyle \frac{1-\lambda }{\lambda }}b-(b-\epsilon )\right)m(G_n)\\ \ge \mathsf{\Phi }\left(v_n-{\displaystyle \frac{1-\lambda }{\lambda }}b\right)m(G_n)-{\displaystyle \frac{\epsilon }{2}}{\mathrm{p}}_{-}(b)m(G_0)\\ =\epsilon {\mathrm{p}}_{-}(b)m(G_0)-{\displaystyle \frac{\epsilon }{2}}{\mathrm{p}}_{-}(b)m(G_0)\\ ={\displaystyle \frac{\epsilon }{2}}{\mathrm{p}}_{-}(b)m(G_0).\end{array}$$

(46)

Using $\mathsf{\Phi }\in {\Delta }_2$ again, there exists ${\epsilon }_0>0$ such that ${\Vert x_n-x_m\Vert }_{\mathsf{\Phi }}\ge {\epsilon }_0,$ a contradiction. □

Corollary 1.

Orlicz space  $L_{\mathsf{\Phi }}(G)$ equipped with the Luxemburg norm is locally nearly uniformly convex if and only if

(1) 

$\mathsf{\Phi }\in {\Delta }_2$;

(2) 

$a(\mathsf{\Phi })=0$;

(3) 

$\mathsf{\Phi }$ is strictly convex on $[0,+\infty )$.

.

Proof of Corollary 1.

By Theorem 3 we get that if $L_{\mathsf{\Phi }}(G)$ is locally nearly uniformly convex, then $\mathsf{\Phi }\in {\Delta }_2$ and $a(\mathsf{\Phi })=0$.

Suppose $\mathsf{\Phi }$ is not strictly convex in ${ℝ}^{+}$. Then there exists an interval $[a,b]\subset {ℝ}^{+}$ such that $\mathsf{\Phi }(u)$ is affine in $[a,b]$. Take $G_0\subset G$ with $0<\mathsf{\Phi }\left({\displaystyle \frac{a+b}{2}}\right)m\left(G_0\right)<1$. Take $c>0$ large enough for which

$$\mathsf{\Phi }(c)m\left(G| G_0\right)+\mathsf{\Phi }\left({\displaystyle \frac{a+b}{2}}\right)m\left(G_0\right)>1.$$

(47)

Using $(G,\Sigma ,\mu )$ is a finite nonatomic measurable space, there is $G_1\subset G| G_0$ such that

$$\mathsf{\Phi }(c)m\left(G_1\right)+\mathsf{\Phi }\left({\displaystyle \frac{a+b}{2}}\right)m\left(G_0\right)=1.$$

(48)

Put

$$x(t)=c{\chi }_{G_1}+{\displaystyle \frac{a+b}{2}}{\chi }_{G_0}.$$

(49)

Then

$$m(\{t\in G:x(t)\in SAI(\mathsf{\Phi })\})\ge m(G_0)>0,$$

(50)

i.e., $x$ is not a locally nearly uniformly convex point. □

Sufficiency. If $\mathsf{\Phi }$ satisfy the conditions (1)–(3), then $L_{\mathsf{\Phi }}(G)$ is locally uniformly convex. Hence, $L_{\mathsf{\Phi }}(G)$ is locally nearly uniformly convex (see [15]).

For completeness, we give a short proof here.

Let $x_n\in S(L_{\mathsf{\Phi }}(G))$ and $x\in S(L_{\mathsf{\Phi }}(G))$ such that

$$\underset{n\to \infty }{\lim }{\Vert x_n+x\Vert }_{\mathsf{\Phi }}=2.$$

(51)

By $\mathsf{\Phi }\in {\Delta }_2$ and $\underset{n\to \infty }{\lim }{‖{\displaystyle \frac{x_n+x}{2}}‖}_{\mathsf{\Phi }}=1$, we have that

$$\underset{n\to \infty }{\lim }{I}_{\mathsf{\Phi }}\left({\displaystyle \frac{x_n+x}{2}}\right)=1.$$

(52)

using $\mathsf{\Phi }$ is strictly convex on ${ℝ}^{+}$ and Lemma 2.26 in [18], we get that

$$x_n\to x \mathrm{in} \mathrm{measure}.$$

(53)

By $\mathsf{\Phi }\in {\Delta }_2$, for any $\epsilon >0$, there exists a ${\epsilon }^{\prime }>0$ such that

$$I_{\mathsf{\Phi }}(z)<{\epsilon }^{\prime },$$

(54)

which implies that

$${\Vert z\Vert }_{\mathsf{\Phi }}<{\epsilon }^{\prime }.$$

(55)

Using $\mathsf{\Phi }\in {\Delta }_2$ again, there exists a $\delta >0$ such that

$$I_{\mathsf{\Phi }}\left(x{\chi }_e\right)={{\displaystyle \int }}_e\mathsf{\Phi }(x\left(t\right))dt<{\displaystyle \frac{{\epsilon }^{\prime }}{2}},$$

(56)

whenever $m(e)<\delta$.

In light of Egrov’s theorem, for $\delta >0$ mentioned above, there exists a measurable subset $e_0$ with $m(e_0)<\delta$ for which

$$x_n(t)\to x(t)\mathrm{uniformly} \mathrm{on} t\in G|e_0.$$

(57)

Hence,

$$I_{\mathsf{\Phi }}\left(x_n{\chi }_{G\left|e_0\right.}\right)\to I_{\mathsf{\Phi }}\left(x{\chi }_{G\left|e_0\right.}\right),$$

(58)

i.e., there is a $m_0\in \mathbb{N}$ such that

$$I_{\mathsf{\Phi }}\left(x_n{\chi }_{G\left|e_0\right.}\right)\ge I_{\mathsf{\Phi }}\left(x{\chi }_{G\left|e_0\right.}\right)-{\displaystyle \frac{{\epsilon }^{\prime }}{2}} \mathrm{and} {\Vert x_n{\chi }_{G\left|e_0\right.}-x{\chi }_{G\left|e_0\right.}\Vert }_{\mathsf{\Phi }}<\epsilon ,$$

(59)

whenever $n\ge m_0$.

Therefore,

$$\begin{array}{l}{I}_{\mathsf{\Phi }}\left(x_n{\chi }_{e_0}\right)=1-I_{\mathsf{\Phi }}\left(x_n{\chi }_{G\left|e_0\right.}\right)\\ \le 1-(I_{\mathsf{\Phi }}\left(x{\chi }_{G\left|e_0\right.}\right)-{\displaystyle \frac{{\epsilon }^{\prime }}{2}})\\ \le I_{\mathsf{\Phi }}\left(x{\chi }_{e_0}\right)+{\displaystyle \frac{{\epsilon }^{\prime }}{2}}\\ \le {\epsilon }^{\prime },\end{array}$$

(60)

i.e.,

$${\Vert x_n{\chi }_{e_0}\Vert }_{\mathsf{\Phi }}<\epsilon \mathrm{for} n\ge n_0.$$

(61)

So

$$\begin{array}{ll}\hfill {\Vert x_n-x\Vert }_{\mathsf{\Phi }}& ={\Vert (x_n-x){\chi }_{G\left|e_0\right.}+x_n{\chi }_{e_0}-x{\chi }_{e_0}\Vert }_{\mathsf{\Phi }}\hfill \\ & \le {\Vert (x_n-x){\chi }_{G\left|e_0\right.}\Vert }_{\mathsf{\Phi }}+{\Vert x_n{\chi }_{e_0}\Vert }_{\mathsf{\Phi }}+{\Vert x{\chi }_{e_0}\Vert }_{\mathsf{\Phi }}\hfill \\ & \le \epsilon +\epsilon +\epsilon \hfill \\ & =3\epsilon ,\hfill \end{array}$$

(62)

whenever $n\ge m_0$, i.e.,

$$\underset{n\to \infty }{\lim }{\Vert x_n-x\Vert }_{\mathsf{\Phi }}=0.$$

(63)

Theorem 4.

A point  $x\in S(l_{\mathsf{\Phi }})$ is $(LNUC)$ point if and only if ① $\mathsf{\Phi }\in {\delta }_2$; ② $\mathsf{\Phi }\in {\delta }_2^{\circ }$ or $\{i\in N:|x(i)|\in S{C}_{\mathsf{\Phi }}^{-}\}=\varphi$.

Proof of Theorem 4.

Necessity. ① Suppose that $\mathsf{\Phi }\in {\delta }_2$. Then, there exists a sequence ${\left\{u_n\right\}}_{n=1}^{\infty }$ such that u_n $\searrow$ 0 and

$$\mathsf{\Phi }(2u_n)\ge 2^{n+1}\mathsf{\Phi }(u_n) \mathrm{for} n=1,2,\cdots$$

(64)

Take $l_n\in \mathbb{N}$ such that

$${\displaystyle \frac{1}{2^{n+1}}}\le l_n\mathsf{\Phi }(u_n)<{\displaystyle \frac{1}{2^n}} \mathrm{for} n=1,2,\cdots$$

(65)

Put $\mathbb{N}=N_1\bigcup N_2,N_1\bigcap N_2=\varphi , x|N_1\in h_{\mathsf{\Phi }},$ $N_1$ is an infinite set.

$$x|N_1=(x_1(1),x_1(2),\cdots ) x|N_2=(x_2(1),x_2(2),\cdots ).$$

For convenience, $x=(x_1(1),x_1(2),\cdots ,x_2(1),x_2(2),\cdots ),$

$$x_n=(x_1(1),x_1(2),\cdots ,x_1(L_{n-1}),\stackrel{{\displaystyle |⟵l_n⟶|}}{u_n,u_n,\cdots ,u_n},0,0,\cdots ,x_2(1),x_2(2),\cdots ),$$

where $L_{n-1}=l_1+l_2+\cdots +l_{n-1}.$

Put $y_n=(0,\cdots ,0,x_1(L_{n-1}+1),\cdots ,x_1(L_n),0,0,\cdots )$. Then $y_n\in h_{\mathsf{\Phi }}$. So we have that

$$\underset{n\to \infty }{\lim }{\Vert y_n\Vert }_{\mathsf{\Phi }}=0 \mathrm{and} \underset{n\to \infty }{\lim }{\Vert x_n\Vert }_{\mathsf{\Phi }}={\Vert x\Vert }_{\mathsf{\Phi }}=1$$

(66)

Put

$$z_n=(\stackrel{\left|⟵l_{n-1}⟶\right|}{0,0,\cdots ,0},{\displaystyle \stackrel{|⟵l_n⟶|}{u_n,u_n,\cdots ,u_n}},0,0\cdots ).$$

(67)

Then

$$I_{\mathsf{\Phi }}\left(z_n\right)=l_n\mathsf{\Phi }\left(u_n\right)\le {\displaystyle \frac{1}{2^n}}\le 1,$$

(68)

and

$$\begin{array}{l}{I}_{\mathsf{\Phi }}\left(2z_n\right)=l_n\mathsf{\Phi }\left(2u_n\right)\ge l_n{2}^{n+1}\mathsf{\Phi }\left(u_n\right)\\ \ge 2^{n+1}\cdot {\displaystyle \frac{1}{2^{n+1}}}=1\end{array}$$

(69)

i.e.,

$${\Vert z_n\Vert }_{\mathsf{\Phi }}\ge {\displaystyle \frac{1}{2}}.$$

(70)

By ${\Vert x_n-x_m\Vert }_{\mathsf{\Phi }}\ge {\Vert z_n\Vert }_{\mathsf{\Phi }}-{\Vert y_n\Vert }_{\mathsf{\Phi }}(n>m)$, there exists a $n_0\in \mathbb{N}$ such that

$$sep(x_n)\ge {\displaystyle \frac{1}{3}} \mathrm{for} n\ge n_0.$$

(71)

For any $\lambda \in (0,1)$, we also have the following

$$\underset{n\to \infty }{\lim }{\Vert \lambda x_n+(1-\lambda )x\Vert }_{\mathsf{\Phi }}=1,$$

(72)

for each $n\in \mathbb{N}$, contradiction.

② Assume that condition (2) does not hold. Then, there is a $i_0\in \mathbb{N}$ such that $x(i_0)\in S{C}_{\mathsf{\Phi }}^{-}$ for convenience we may assume that $x(i)\ge 0$ and for any $\lambda \in (0,1)$ there exists a sequence $\left\{v_n\right\}$ with $v_n\searrow 0$ and

$$\mathsf{\Phi }(\lambda v_n)\ge \lambda \left(1-{\displaystyle \frac{1}{2^n}}\right)\mathsf{\Phi }(v_n) \mathrm{for} n=1,2,\cdots$$

(73)

Using $x(i_0)\in S{C}_{\mathsf{\Phi }}^{-}$, for convenience, we may assume that $x(1)\in S{C}_{\mathsf{\Phi }}^{-}$, i.e., there is a $s>0$ for which

$$\mathsf{\Phi }(u) \mathrm{is} \mathrm{affine} \mathrm{on} [x(1)-s,x(1)].$$

(74)

i.e., there are $A,B\in ℝ$ such that

$$\mathsf{\Phi }(u)=Au+B, u\in [x(1)-s,x(1)].$$

(75)

Take $l_n\in \mathbb{N}$ such that

$$As-{\displaystyle \frac{1}{2^n}}\le l_n\mathsf{\Phi }(u_n)\le As \mathrm{for} n=1,2,\cdots$$

(76)

Define

$$\begin{array}{l}x=(x(1)-s,\stackrel{{\displaystyle |⟵l_1⟶|}}{v_1,v_1,\cdots ,v_1},0,0\cdots )\\ x_2=(x(1)-s,\stackrel{{\displaystyle |⟵l_1⟶|}}{x(2),0,\cdots ,0},\stackrel{{\displaystyle |⟵l_2⟶|}}{v_2,v_2,\cdots ,v_2,}0,0\cdots ).\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \end{array}$$

(77)

Then,

$$\begin{array}{ll}\hfill I_{\mathsf{\Phi }}(x_n)=& \mathsf{\Phi }(x(1)-s)+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})+l_n\mathsf{\Phi }(v_n)\hfill \\ & \le A(x(1)-s)+B+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})+AS\hfill \\ & =Ax(1)+B+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})\hfill \\ & \le I_{\mathsf{\Phi }}(x)=1,\hfill \end{array}$$

(78)

and

$$\begin{array}{l}1\ge I_{\mathsf{\Phi }}\left({\displaystyle \frac{x_n+x}{2}}\right)=\mathsf{\Phi }\left(x(1)-{\displaystyle \frac{s}{2}}\right)+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})+\lambda (x(1)-s)\\ +(1-\lambda )(x(1)-s)+(1-\lambda )(x(1)-s)\\ =x(1)-\lambda s.\end{array}$$

(79)

Hence,

$$\begin{array}{ll}\hfill 1& \ge I_{\mathsf{\Phi }}(\lambda x_n+(1-\lambda )x)\hfill \\ & =\mathsf{\Phi }(x(1)-\lambda s)+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})+l_n\mathsf{\Phi }(v_n)\hfill \\ & \ge A(x(1)-s)+B+{\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})+\lambda l_n\left(1-{\displaystyle \frac{1}{2^n}}\right)\mathsf{\Phi }(v_n)\hfill \\ & ={\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})-A\lambda s+\lambda l_n\mathsf{\Phi }(v_n)-{\displaystyle \frac{\lambda }{2^n}}{l}_n\mathsf{\Phi }(v_n).\hfill \\ & \ge {\displaystyle \sum _{i=2}^n\mathsf{\Phi }(x(i)})-{\displaystyle \frac{\lambda }{2^n}}As\hfill \\ & \to I_{\mathsf{\Phi }}(x)=1,\hfill \end{array}$$

(80)

By $\mathsf{\Phi }\in {\delta }_2$, we get that

$$\underset{n\to \infty }{\lim }{\Vert \lambda x_n+(1-\lambda )x\Vert }_{\mathsf{\Phi }}=1.$$

(81)

However, we have for any $m,n\in \mathbb{N}$$(m>n)$

$$I_{\mathsf{\Phi }}(x_m-x_n)\ge l_m\mathsf{\Phi }(v_m)\ge As-{\displaystyle \frac{1}{2^m}}.$$

(82)

Take $n_0$ large enough, for which

$${\displaystyle \frac{1}{2^n}}\le {\displaystyle \frac{1}{2}}As.$$

(83)

Hence,

$$sep(x_n)\ge \min \left\{1,{\displaystyle \frac{1}{2}}As\right\} \mathrm{for} n\ge n_0.$$

(84)

Therefore, for ${\epsilon }_0=\min \{1,{\displaystyle \frac{1}{2}}As\}$, there exists a $\delta \in (0,1)$, $\lambda \in (0,1)$ and $n_1\ge n_0$ such that

$${\Vert \lambda x_n+(1-\lambda )x\Vert }_{\mathsf{\Phi }}<1-\delta ,$$

(85)

contradiction. □

Sufficiency. If $\mathsf{\Phi }\in {\delta }_2\bigcap {\delta }_2^0$, then $l_{\mathsf{\Phi }}$ is $(NUC)$ (see [14]).

In the assumption that $\mathsf{\Phi }\in {\delta }_2$ and $\{i\in \mathbb{N}:|x(i)|\in S{C}_{\mathsf{\Phi }}^{-}\}=\varphi$, from the Proof of Theorem 3 (see [18]), we have

$$\underset{n\to \infty }{\lim }{\Vert x_n-x\Vert }_{\mathsf{\Phi }}=0.$$

(86)

This contradict with ${\Vert x_n-x\Vert }_{\mathsf{\Phi }}\ge \epsilon$.

Corollary 2.

$l_{\mathsf{\Phi }}$ is $(LNUC)$ if and only if ① $\mathsf{\Phi }\in {\delta }_2$; ② $\mathsf{\Phi }\in {\delta }_2^{\circ }$ or $\mathsf{\Phi }$ is strictly convex on ${(0, \mathsf{\Phi }}^{-1}(1)]$.

4. Discussion

Our study focuses on new classes of geometric points, and we study them in Banach spaces and Orlicz sequence spaces with two definitions, three theorems, and a corollary. These points indicate that they play a very significant role in some recent trends related to the theory of Banach and Orlicz spaces.

5. Conclusions

In this study, we introduce the nearly convex point and the locally nearly uniformly convex point. We study these points using the geometric theory of Banach spaces and Orlicz function spaces. First, we provide the general characterization of a nearly convex point in Banach space. Second, the relationship between a locally nearly uniformly convex point and the nearly extreme point is also investigated in Banach space. Furthermore, we characterize locally nearly convex points in Orlicz function spaces and Orlicz sequence spaces equipped with Luxemburg norm.