On geodesic strongly E-convex sets and geodesic strongly E-convex functions

Abstract

In this article, geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold are extended to the so-called geodesic strongly E-convex sets and geodesic strongly E-convex functions. Some properties of geodesic strongly E-convex sets are also discussed. The results obtained in this article may inspire future research in convex analysis and related optimization fields.

1 Introduction

Convexity and its generalizations play an important role in optimization theory, convex analysis, Minkowski space, and fractal mathematics [1–7]. In order to extend the validity of their results to large classes of optimization, these concepts have been generalized and extended in several directions using novel and innovative techniques. Youness [8] defined E-convex sets and E-convex functions, which have some important applications in various branches of mathematical sciences [9–11]. However, some results given by Youness [8] seem to be incorrect according to Yang [12]. Chen [13] extended E-convexity to a semi-E-convexity and discussed some of there properties. Also, Youness and Emam [14] discussed a new class functions which is called strongly E-convex functions by taking the images of two points $x_{1}$ and $x_{2}$ under an operator $E\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ besides the two points themselves. Strong E-convexity was extended to a semi-strong E-convexity as well as quasi- and pseudo-semi-strong E-convexity in [15]. The authors investigated the characterization of efficient solutions for multi-objective programming problems involving semi-strong E-convexity [16].

A generalization of convexity on Riemannian manifolds was proposed by Rapcsak [17] and Udriste [18]. Moreover, Iqbal et al. [19] introduced geodesic E-convex sets and geodesic E-convex functions on Riemannian manifolds.

Motivated by earlier research works [18, 20–25] and by the importance of the concepts of convexity and generalized convexity, we discuss a new class of sets on Riemannian manifolds and a new class of functions defined on them, which are called geodesic strongly E-convex sets and geodesic strongly E-convex functions, and some of their properties are presented.

2 Preliminaries

In this section, we introduce some definitions and well-known results of Riemannian manifolds, which help us throughout the article. We refer to [18] for the standard material on differential geometry.

Let N be a $C^{\infty}$ m-dimensional Riemannian manifold, and $T_{z}N$ be the tangent space to N at z. Also, assume that $\mu_{z}(x_{1},x_{2})$ is a positive inner product on the tangent space $T_{z}N$ ($x_{1},x_{2}\in T_{z}N$), which is given for each point of N. Then a $C^{\infty}$ map $\mu\colon z\rightarrow\mu_{z}$, which assigns a positive inner product $\mu _{z}$ to $T_{z}N$ for each point z of N is called a Riemannian metric.

The length of a piecewise $C^{1}$ curve $\eta\colon [a_{1},a_{2}]\rightarrow N$ which is defined as follows:

$$L(\eta)= \int_{a_{1}}^{a_{2}} \bigl\Vert \acute{ \eta}(x)\bigr\Vert \, dx. $$

We define $d(z_{1},z_{2})= \inf \lbrace L(\eta)\colon\eta\mbox{ is a piecewise } C^{1} \mbox{ curve joining } z_{1} \mbox{ to } z_{2} \rbrace$ for any points $z_{1},z_{2}\in N$. Then d is a distance which induces the original topology on N. As we know on every Riemannian manifold there is a unique determined Riemannian connection, called a Levi-Civita connection, denoted by $\bigtriangledown_{X}Y$, for any vector fields $X,Y\in N$. Also, a smooth path η is a geodesic if and only if its tangent vector is a parallel vector field along the path η, i.e., η satisfies the equation $\bigtriangledown_{\acute{\eta}(t)}\acute{\eta}(t)=0$. Any path η joining $z_{1}$ and $z_{2}$ in N such that $L(\eta )=d(z_{1},z_{2})$ is a geodesic and is called a minimal geodesic.

Finally, assume that $(N,\eta)$ is a complete m-dimensional Riemannian manifold with Riemannian connection ▽. Let $x_{1} , x_{2} \in N$ and $\eta\colon[0,1]\rightarrow N$ be a geodesic joining the points $x_{1}$ and $x_{2}$, which means that $\eta_{x_{1},x_{2}}(0)=x_{2}$ and $\eta_{x_{1},x_{2}}(1)=x_{1}$.

Definition 2.1

[18]

A set B in a Riemannian manifold N is called totally convex if B contains every geodesic $\eta_{x_{1},x_{2}}$ of N whose endpoints $x_{1}$ and $x_{2}$ belong to B.

Note the whole of the manifold N is totally convex, and conventionally, so is the empty set. The minimal circle in a hyperboloid is totally convex, but a single point is not. Also, any proper subset of a sphere is not necessarily totally convex.

The following theorem was proved in [18].

Theorem 2.2

[18]

The intersection of any number of a totally convex sets is totally convex.

Remark 2.3

In general, the union of a totally convex set is not necessarily totally convex.

Definition 2.4

[18]

A function $f\colon B\rightarrow\mathbb{R}$ is called a geodesic convex function on a totally convex set $B\subset N$ if for every geodesic $\eta_{x_{1},x_{2}}$, then

$$f\bigl(\eta_{x_{1},x_{2}}(\gamma)\bigr)\leq\gamma f(x_{1})+(1- \gamma)f(x_{2}) $$

holds for all $x_{1},x_{2}\in B$ and $\gamma\in[0,1]$.

In 2005, strongly E-convex sets and strongly E-convex functions were introduced by Youness and Emam [14] as follows.

Definition 2.5

[14]

• (1) A subset $B\subseteq\mathbb{R}^{n}$ is called a strongly E-convex set if there is a map $E\colon\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ such that

  $$\gamma\bigl(\alpha b_{1}+E(b_{1})\bigr)+(1-\gamma) \bigl( \alpha b_{2}+E(b_{2})\bigr)\in B $$

  for each $b_{1},b_{2}\in B$, $\alpha\in[0,1]$ and $\gamma\in[0,1]$.

• (2) A function $f\colon B\subseteq\mathbb {R}^{n}\rightarrow\mathbb{R}$ is called a strongly E-convex function on N if there is a map $E\colon\mathbb {R}^{n}\rightarrow\mathbb{R}^{n}$ such that B is a strongly E-convex set and

  $$f\bigl(\gamma\bigl(\alpha b_{1}+E(b_{1})\bigr)+(1-\gamma) \bigl(\alpha b_{2}+E(b_{2})\bigr)\bigr)\leq \gamma f \bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$

  for each $b_{1},b_{2}\in B$, $\alpha\in[0,1]$ and $\gamma\in[0,1]$.

In 2012, the geodesic E-convex set and geodesic E-convex functions on a Riemannian manifold were introduced by Iqbal et al. [19] as follows.

Definition 2.6

[19]

• (1) Assume that $E\colon N\rightarrow N$ is a map. A subset B in a Riemannian manifold N is called geodesic E-convex iff there exists a unique geodesic $\eta_{E(b_{1}),E(b_{2})}(\gamma)$ of length $d(b_{1},b_{2})$, which belongs to B, for each $b_{1},b_{2}\in B$ and $\gamma\in[0,1]$.

• (2) A function $f\colon B\subseteq N \rightarrow\mathbb {R}$ is called geodesic E-convex on a geodesic E-convex set B if

  $$f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr)\leq\gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$

  for all $b_{1},b_{2}\in B$ and $\gamma\in[0,1]$.

3 Geodesic strongly E-convex sets and geodesic strongly E-convex functions

In this section, we introduce a geodesic strongly E-convex (GSEC) set and a geodesic strongly E-convex (GSEC) function in a Riemannian manifold N and discuss some of their properties.

Definition 3.1

Assume that $E\colon N\rightarrow N$ is a map. A subset B in a Riemannian manifold N is called GSEC if and only if there is a unique geodesic $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)$ of length $d(b_{1},b_{2})$, which belongs to B, $\forall b_{1},b_{2}\in B$, $\alpha\in[0,1]$, and $\gamma\in [0,1]$.

Remark 3.2

• (1) Every GSEC set is a GEC set when $\alpha=0$.

• (2) A GEC set is not necessarily a GSEC set. The following example shows this statement.

Example 3.3

Let $N^{2}$ be a 2-dimensional simply complete Riemannian manifold of non-positive sectional curvature, and $B\subset N^{2}$ be an open star-shaped. Let $E\colon N^{2}\rightarrow N^{2}$ be a map such that $E(z)= \lbrace y\colon y\in \operatorname{ker}(B), \forall z\in B \rbrace$. Then B is GEC; on the other hand it is not GSEC.

Proposition 3.4

Every convex set $B\subset N$ is a GSEC set.

Proof

Let us take a map $E\colon N\rightarrow N$ such as $E=I$ where I is the identity map and $\alpha=0$, then we have the required result. □

Note if we take the mapping $E(x)=(1-\alpha)x$, $x\in B$, then the definition of a GSE reduces to the definition of a t-convex set.

Theorem 3.5

If $B\subset N$ is a GSEC set, then $E(B)\subseteq B$.

Proof

Since B is a GSEC set, we have for each $b_{1},b_{2}\in B$, $\alpha \in[0,1]$, and $\gamma\in[0,1]$,

$$\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in B. $$

For $\gamma=0$ and $\alpha=0$, we have $\eta _{E(b_{1}),E(b_{2})}(0)=E(b_{2})\in B$, then $E(B)\subseteq B$. □

Theorem 3.6

If $\lbrace B_{j}, j\in I \rbrace$ is an arbitrary family of GSEC subsets of N with respect to the mapping $E\colon N\rightarrow N$, then the intersection $\bigcap_{j\in I}B_{j}$ is a GSEC subset of N.

Proof

If $\bigcap_{j\in I}B_{j}$ is an empty set, then it is obviously a GSEC subset of N. Assume that $b_{1},b_{2}\in\bigcap_{j\in I} B_{j}$, then $b_{1},b_{2} \in B_{j}$, $\forall j\in I$. By the GSEC of $B_{j}$, we get $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma )\in B_{j}$, $\forall j\in I$, $\alpha\in[0,1]$, and $\gamma\in[0,1]$. Hence, $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in \bigcap_{j\in I} B_{j}$, $\forall\alpha\in[0,1]$ and $\gamma\in[0,1]$. □

Remark 3.7

The above theorem is not generally true for the union of GSEC subsets of N.

Now, we extend the definition of a GEC function on a Riemannian manifold to a GSEC function on a Riemannian manifold.

Definition 3.8

A real-valued function $f\colon B\subset N\rightarrow\mathbb{R}$ is said to be a GSEC function on a GSEC set B, if

$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr), $$

$\forall b_{1},b_{2}\in B$, $\alpha\in[0,1]$, and $\gamma\in[0,1]$. If the above inequality is strict for all $b_{1},b_{2}\in B$, $\alpha b_{1}+E(b_{1})\neq\alpha b_{2}+E(b_{2})$, $\alpha\in[0,1]$, and $\gamma \in(0,1)$, then f is called a strictly GSEC function.

Remark 3.9

• (1) Every GSEC function is a GEC function when $\alpha=0$. The following example shows that a GEC function is not necessarily a GSEC function.

Example 3.10

Consider the function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ where $f(b)= -|b|$ and suppose that $E\colon\mathbb {R}\rightarrow\mathbb{R}$ is given as $E(b)=-b$. We consider the geodesic η such that

$$\begin{aligned} \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) =& \textstyle\begin{cases} - [\alpha b_{2}+E(b_{2}) +\gamma(\alpha b_{1}+E(b_{1})-\alpha b_{2}-E(b_{2})) ] ;& b_{1}b_{2}\geq0, \\ - [\alpha b_{2}+E(b_{2}) +\gamma(\alpha b_{2}+E(b_{2})-\alpha b_{1}-E(b_{1})) ] ;& b_{1}b_{2}< 0 \end{cases}\displaystyle \\ =& \textstyle\begin{cases} - [(\alpha-1) b_{2} +\gamma((\alpha-1) b_{1}+(1-\alpha) b_{2}) ] ;& b_{1}b_{2}\geq0, \\ - [(\alpha-1) b_{2} +\gamma ((\alpha-1) b_{2}+(1-\alpha) b_{1}) ] ;& b_{1}b_{2}< 0. \end{cases}\displaystyle \end{aligned}$$

If $\alpha=0$, then

$$ \eta_{E(b_{1}),E(b_{2})}(\gamma) = \textstyle\begin{cases} {[ b_{2} +\gamma( b_{1}-b_{2}) ]} ;& b_{1}b_{2}\geq0, \\ {[ b_{2} +\gamma( b_{2}- b_{1}) ]} ;& b_{1}b_{2}< 0. \end{cases} $$

If $b_{1}, b_{2}\geq0$, then

$$\begin{aligned} f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr) =& f\bigl(b_{2}+\gamma (b_{1}-b_{2})\bigr) \\ =& -\bigl[(1-\gamma)b_{2}+\gamma b_{1}\bigr]. \end{aligned}$$

On the other hand

$$ \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr)= \gamma f(-b_{1})+(1-\gamma )f(-b_{2}) = -\bigl[(1-\gamma)b_{2}+\gamma b_{1}\bigr]. $$

Hence, $f(\eta_{E(b_{1}),E(b_{2})}(\gamma))\leq\gamma f(E(b_{1}))+(1-\gamma)f(E(b_{2}))$, $\forall\gamma\in[0,1]$.

Similarly, the above inequality holds true when $b_{1},b_{2}<0$.

Now, let $b_{1}<0$, $b_{2}>0$, then

$$\begin{aligned} f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr) =& f\bigl(b_{2}+ \gamma(b_{2}-b_{1})\bigr) \\ =& -\bigl[(1+\gamma)b_{2}-\gamma b_{1}\bigr]. \end{aligned}$$

On the other hand

$$ \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) = \gamma f(-b_{1})+(1-\gamma )f(-b_{2}) = \gamma b_{1}-(1-\gamma)b_{2}. $$

It follows that

$$f\bigl(\eta_{E(b_{1}),E(b_{2})}(\gamma)\bigr)\leq\gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma )f\bigl(E(b_{2})\bigr) $$

if and only if

$$-\bigl[(1+\gamma)b_{2}-\gamma b_{1}\bigr]\leq\gamma b_{1}-(1-\gamma)b_{2} $$

if and only if

$$-2\gamma b_{2}\leq0, $$

which is always true for all $\gamma\in[0,1]$.

Similarly, $f(\eta_{E(b_{1}),E(b_{2})}(\gamma))\leq\gamma f(E(b_{1}))+(1-\gamma)f(E(b_{2}))$, $\forall\gamma\in[0,1]$ also holds for $b_{1}>0$ and $b_{2}<0$.

Thus, f is a GEC function on $\mathbb{R}$, but it is not a GSEC function because if we take $b_{1}=0$, $b_{2}=-1$ and $\gamma=\frac {1}{2}$, then

$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) =& f\biggl(\frac{1}{2}\alpha- \frac{1}{2}\biggr) \\ =& \frac{1}{2}\alpha-\frac {1}{2} \\ > & \frac{1}{2}f\bigl(E(0)\bigr)+\frac{1}{2}f\bigl(E(-1)\bigr) \\ =& \frac{-1}{2} ,\quad \forall\alpha\in ( 0,1 ] . \end{aligned}$$

• (2) Every g-convex function f on a convex set B is a GSEC function when $\alpha=0$ and E is the identity map.

Proposition 3.11

Assume that $f\colon B\rightarrow\mathbb{R}$ is a GSEC function on a GSEC set $B\subseteq N$, then $f(\alpha b+E(b))\leq f(E(b))$, $\forall b\in B$ and $\alpha\in[0,1]$.

Proof

Since $f\colon B\rightarrow\mathbb{R}$ is a GSEC function on a GSEC set $B\subseteq N$, then $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \in B$, $\forall b_{1},b_{2}\in B$, $\alpha\in [0,1]$, and $\gamma\in[0,1]$. Also,

$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) $$

thus, for $\gamma=1$, we get $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)=\alpha b_{1}+E(b_{1})$. Then

$$f\bigl(\alpha b_{1}+E(b_{1})\bigr)\leq f \bigl(E(b_{1})\bigr) . $$

 □

Theorem 3.12

Consider that $B\subseteq N$ is a GSEC set and $f_{1}\colon B\rightarrow\mathbb{R}$ is a GSEC function. If $f_{2}\colon I\rightarrow\mathbb{R}$ is a non-decreasing convex function such that $\operatorname{rang}(f_{1})\subset I$, then $f_{2}\circ f_{1}$ is a GSEC function on B.

Proof

Since $f_{1}$ is a GSEC function, for all $b_{1},b_{2}\in B$, $\alpha \in[0,1]$, and $\gamma\in[0,1]$,

$$f_{1}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{1}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{1} \bigl(E(b_{2})\bigr). $$

Since $f_{2}$ is a non-decreasing convex function,

$$\begin{aligned}& f_{2}\circ f_{2}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma )\bigr) \\& \quad = f_{2} \bigl( f_{2}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma)\bigr) \bigr) \\& \quad \leq f_{2} \bigl(\gamma f_{1} \bigl(E(b_{1}) \bigr)+(1-\gamma) f_{1}\bigl(E(b_{2})\bigr) \bigr) \\& \quad \leq \gamma f_{2} \bigl( f_{1} \bigl(E(b_{1}) \bigr) \bigr) +(1-\gamma) f_{2} \bigl( f_{1} \bigl(E(b_{2})\bigr) \bigr) \\& \quad = \gamma(f_{2}\circ f_{1}) \bigl(E(b_{1})\bigr) +(1-\gamma) (f_{2}\circ f_{1}) \bigl(E(b_{2}) \bigr), \end{aligned}$$

which means that $f_{2}\circ f_{1}$ is a GSEC function on B. Similarly, if $f_{2}$ is a strictly non-decreasing convex function, then $f_{2}\circ f_{1}$ is a strictly GSEC function. □

Theorem 3.13

Assume that $B\subseteq N$ is a GSEC set and $f_{j}\colon B\rightarrow\mathbb{R}$, $j=1,2,\ldots,m$ are GSEC functions. Then the function

$$f=\sum_{j=1}^{m}n_{j}f_{j} $$

is GSEC on B, $\forall n_{j}\in\mathbb{R}$, $n_{j}\geq0$.

Proof

Since $f_{j}$, $j=1,2,\ldots,m$ are GSEC functions, $\forall b_{1},b_{2}\in B$, $\alpha\in[0,1]$, and $\gamma\in[0,1]$, we have

$$f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{j} \bigl(E(b_{2})\bigr). $$

It follows that

$$n_{j}f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma )\bigr)\leq \gamma n_{j} f_{j}\bigl(E(b_{1})\bigr)+(1- \gamma)n_{j}f_{j}\bigl(E(b_{2})\bigr). $$

Then

$$\begin{aligned}& \sum_{j=1}^{m}n_{j}f_{j} \bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})} (\gamma)\bigr) \\& \quad \leq \gamma\sum _{j=1}^{m} n_{j} f_{j} \bigl(E(b_{1})\bigr)+(1-\gamma)\sum_{j=1}^{m}n_{j}f_{j} \bigl(E(b_{2})\bigr) \\& \quad = \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$

Thus, f is a GSEC function. □

Theorem 3.14

Let $B\subseteq N$ be a GSEC set and $\lbrace f_{j},j\in I \rbrace$ be a family of real-valued functions defined on B such that $\sup_{j\in I}f_{j}(b)$ exists in $\mathbb{R}$, $\forall b\in B$. If $f_{j}\colon B\rightarrow\mathbb{R}$, $j\in I$ are GSEC functions on B, then the function $f\colon B\rightarrow \mathbb{R}$, defined by $f(b)=\sup_{j\in I}f_{j}(b)$, $\forall b\in B$ is GSEC on B.

Proof

Since $f_{j}$, $j\in I$ are GSEC functions on a GSEC set B, $\forall b_{1},b_{2}\in B$, $\alpha\in[0,1]$, and $\gamma\in[0,1]$, we have

$$f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)f_{j} \bigl(E(b_{2})\bigr). $$

Then

$$\begin{aligned}& \sup_{j\in I}f_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \bigr) \\& \quad \leq \sup_{j\in I} \bigl[ \gamma f_{j} \bigl(E(b_{1})\bigr)+(1-\gamma)f_{j}\bigl(E(b_{2}) \bigr) \bigr] \\& \quad = \gamma\sup_{j\in I} f_{j}\bigl(E(b_{1}) \bigr)+(1-\gamma)\sup_{j\in I} f_{j} \bigl(E(b_{2})\bigr) \\& \quad = \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$

Hence,

$$f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr)\leq \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr), $$

which means that f is a GSEC function on B. □

Proposition 3.15

Assume that $h_{j}\colon N\rightarrow\mathbb{R}$, $j=1,2,\ldots,m$ are GSEC functions on N, with respect to $E\colon N\rightarrow N$. If $E(B)\subseteq B$, then $B= \lbrace b\in N\colon h_{j}(b)\leq0, j=1,2,\ldots,m \rbrace$ is a GSEC set.

Proof

Since $h_{j}$, $j=1,2,\ldots m$ are GSEC functions,

$$\begin{aligned} h_{j}\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq & \gamma h_{j}\bigl(E(b_{1})\bigr)+(1-\gamma)h_{j} \bigl(E(b_{2})\bigr) \\ \leq& 0, \end{aligned}$$

$\forall b_{1},b_{2}\in B$, $\alpha\in[0,1]$, and $\gamma\in[0,1]$. Since $E(B) \subseteq B$, $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma) \in B$. Hence, B is a GSEC set. □

4 Epigraphs

Youness and Emam [14] defined a strongly $E\times F$-convex set where $E\colon\mathbb{R}^{n} \rightarrow\mathbb{R}^{n}$ and $F\colon\mathbb{R} \rightarrow\mathbb{R}$ and studied some of its properties. In this section, we generalize a strongly $E\times F$-convex set to a geodesic strongly $E\times F$-convex set on Riemannian manifolds and discuss GSEC functions in terms of their epigraphs. Furthermore, some properties of GSE sets are given.

Definition 4.1

Let $B\subset N\times\mathbb{R}$, $E\colon N\rightarrow N$ and $F\colon\mathbb{R} \rightarrow\mathbb{R}$. A set B is called geodesic strongly $E\times F$-convex if $(b_{1},\beta _{1}),(b_{2},\beta_{2})\in B$ implies

$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(\beta_{1})+(1-\gamma)F( \beta_{2}) \bigr) \in B $$

for all $\alpha\in[0,1]$ and $\gamma\in[0,1]$.

It is not difficult to prove that $B\subseteq N$ is a GSEC set if and only if $B\times\mathbb{R}$ is a geodesic strongly $E\times F$-convex set.

An epigraph of f is given by

$$\operatorname{epi}(f)= \bigl\lbrace (b,a)\colon b\in B, a\in\mathbb{R}, f(b)\leq a \bigr\rbrace . $$

A characterization of a GSEC function in terms of its $\operatorname{epi}(f)$ is given by the following theorem.

Theorem 4.2

Let $E\colon N\rightarrow N$ be a map, $B\subseteq N$ be a GSEC set, $f\colon B\rightarrow\mathbb{R}$ be a real-valued function and $F\colon\mathbb{R}\rightarrow\mathbb{R}$ be a map such that $F(f(b)+a)=f(E(b))+a$, for each non-negative real number a. Then f is a GSEC function on B if and only if $\operatorname{epi}(f)$ is geodesic strongly $E\times F$-convex on $B\times\mathbb{R}$.

Proof

Assume that $(b_{1},a_{1}) ,(b_{2},a_{2})\in \operatorname{epi}(f)$. If B is a GSEC set, then $\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\in B$, $\forall\alpha\in[0,1]$ and $\gamma\in [0,1]$. Since $E(b_{1})\in B$ for $\alpha=0$, $\gamma=1$, also $E(b_{2})\in B$ for $\alpha=0$, $\gamma=0$, let $F(a_{1})$ and $F(a_{2})$ be such that $f(E(b_{1}))\leq F(a_{1})$ and $f(E(b_{2}))\leq F(a_{2})$. Then $(E(b_{1}),F(a_{1})),(E(b_{2}),F(a_{2}))\in \operatorname{epi}(f)$.

Let f be GSEC on B, then

$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq& \gamma f\bigl(E(b_{1}) \bigr)+(1-\gamma)f\bigl(E(b_{2})\bigr) \\ \leq& \gamma F(a_{1})+(1-\gamma)F(a_{2}). \end{aligned}$$

Thus, $( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1-\gamma)F(a_{2}) ) \in \operatorname{epi}(f)$, then $\operatorname{epi}(f)$ is geodesic strongly $E\times F$-convex on $B\times\mathbb{R}$.

Conversely, assume that $\operatorname{epi}(f)$ is geodesic strongly $E\times F$-convex on $B\times\mathbb{R}$. Let $b_{1},b_{2}\in B$, $\alpha\in [0,1]$, and $\gamma\in[0,1]$, then $(b_{1},f(b_{1}))\in \operatorname{epi}(f)$ and $(b_{2},f(b_{2}))\in \operatorname{epi}(f)$. Now, since $\operatorname{epi}(f)$ is geodesic strongly $E\times F$-convex on $B\times\mathbb{R}$, we obtain $( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(f(b_{1}))+(1-\gamma)F(f(b_{2})) ) \in \operatorname{epi}(f)$, then

$$\begin{aligned} f\bigl(\eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma)\bigr) \leq& \gamma F\bigl(f(b_{1}) \bigr)+(1-\gamma)F\bigl(f(b_{2})\bigr) \\ =& \gamma f\bigl(E(b_{1})\bigr)+(1-\gamma)f\bigl(E(b_{2}) \bigr). \end{aligned}$$

This shows that f is a GSEC function on B. □

Theorem 4.3

Assume that $\lbrace B_{j}, j\in I \rbrace$ is a family of geodesic strongly $E\times F$-convex sets. Then the intersection $\bigcap_{j\in I}B_{j}$ is a geodesic strongly $E\times F$-convex set.

Proof

Assume that $(b_{1},a_{1}) ,(b_{2},a_{2})\in\bigcap_{j\in I}B_{j}$, so $\forall j\in I$, $(b_{1},a_{1}) ,(b_{2},a_{2})\in B_{j}$. Since $B_{j}$ is the geodesic strongly $E\times F$-convex sets $\forall j\in I$, we have

$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1- \gamma)F(a_{2}) \bigr)\in B_{j} , $$

$\forall\alpha\in[0,1]$ and $\gamma\in[0,1]$. Therefore,

$$\bigl( \eta_{\alpha b_{1}+E(b_{1}),\alpha b_{2}+E(b_{2})}(\gamma ),\gamma F(a_{1})+(1- \gamma)F(a_{2}) \bigr)\in\bigcap_{j\in I}B_{j}, $$

$\forall\alpha\in[0,1]$ and $\gamma\in[0,1]$. Then $\bigcap_{j\in I}B_{j}$ is a geodesic strongly $E\times F$-convex set. □

Theorem 4.4

Assume that $E\colon N \rightarrow N$ and $F\colon\mathbb {R}\rightarrow\mathbb{R}$ are two maps such that $F(f(b)+a)=f(E(b))+a$ for each non-negative real number a. Suppose that $\lbrace f_{j}, j\in I \rbrace$ is a family of real-valued functions defined on a GSEC set $B\subseteq N$ which are bounded from above. If $\operatorname{epi}(f_{j})$ are geodesic strongly $E\times F$-convex sets, then the function f which is given by $f(b)=\sup_{j\in I}f_{j}(b)$, $\forall b\in B$, is a GSEC function on B.

Proof

If each $f_{j}$, $j\in I$ is a GSEC function on a GSEC geodesic set B, then

$$\operatorname{epi}(f_{j})= \bigl\lbrace (b,a)\colon b\in B, a\in \mathbb{R}, f_{j}(b)\leq a \bigr\rbrace $$

are geodesic strongly $E\times F$-convex on $B\times\mathbb{R}$. Therefore,

$$\begin{aligned} \bigcap_{j\in I}\operatorname{epi}(f_{j}) =& \bigl\lbrace (b,a)\colon b\in B, a\in \mathbb{R}, f_{j}(b) \leq a, j\in I \bigr\rbrace \\ =& \bigl\lbrace (b,a)\colon b\in B, a\in\mathbb{R}, f(b)\leq a \bigr\rbrace \end{aligned}$$

is geodesic strongly $E\times F$-convex set. Then, according to Theorem 4.2 we see that f is a GSEC function on B. □