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. (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. (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. (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. (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. (1)

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

  2. (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. (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

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