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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on deriving the results and approved the final manuscript.
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.
Anzeige
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:
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.
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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 \(\eta_{x_{1},x_{2}}(1)=x_{1} \).
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.
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
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.
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
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 ofNwith respect to the mapping\(E\colon N\rightarrow N \), then the intersection\(\bigcap_{j\in I}B_{j} \)is a GSEC subset ofN.
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
\(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), 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
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
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,
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 onB.
Proof
Since \(f_{1} \) is a GSEC function, for all \(b_{1},b_{2}\in B\), \(\alpha \in[0,1] \), and \(\gamma\in[0,1] \),
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 onB, \(\forall n_{j}\in\mathbb{R}\), \(n_{j}\geq0 \).
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
Let\(B\subseteq N \)be a GSEC set and\(\lbrace f_{j},j\in I \rbrace\)be a family of real-valued functions defined onBsuch 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 onB, 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 onB.
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
\(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), 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
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.
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 numbera. Thenfis a GSEC function onBif 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 \((E(b_{1}),F(a_{1})),(E(b_{2}),F(a_{2}))\in \operatorname{epi}(f) \).
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
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
\(\forall\alpha\in[0,1]\) 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 numbera. 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 functionfwhich is given by\(f(b)=\sup_{j\in I}f_{j}(b)\), \(\forall b\in B \), is a GSEC function onB.
Proof
If each \(f_{j}\), \(j\in I \) is a GSEC function on a GSEC geodesic set B, then
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. □
Acknowledgements
The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on deriving the results and approved the final manuscript.