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On geodesic strongly Econvex sets and geodesic strongly Econvex functions
Journal of Inequalities and Applications volume 2015, Article number: 297 (2015)
Abstract
In this article, geodesic Econvex sets and geodesic Econvex functions on a Riemannian manifold are extended to the socalled geodesic strongly Econvex sets and geodesic strongly Econvex functions. Some properties of geodesic strongly Econvex 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 Econvex sets and Econvex 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 Econvexity to a semiEconvexity and discussed some of there properties. Also, Youness and Emam [14] discussed a new class functions which is called strongly Econvex 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 Econvexity was extended to a semistrong Econvexity as well as quasi and pseudosemistrong Econvexity in [15]. The authors investigated the characterization of efficient solutions for multiobjective programming problems involving semistrong Econvexity [16].
A generalization of convexity on Riemannian manifolds was proposed by Rapcsak [17] and Udriste [18]. Moreover, Iqbal et al. [19] introduced geodesic Econvex sets and geodesic Econvex 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 Econvex sets and geodesic strongly Econvex functions, and some of their properties are presented.
2 Preliminaries
In this section, we introduce some definitions and wellknown 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 \(C^{\infty} \) mdimensional 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 LeviCivita 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 mdimensional 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
holds for all \(x_{1},x_{2}\in B \) and \(\gamma\in[0,1] \).
In 2005, strongly Econvex sets and strongly Econvex 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 Econvex 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 Econvex function on N if there is a map \(E\colon\mathbb {R}^{n}\rightarrow\mathbb{R}^{n} \) such that B is a strongly Econvex 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 Econvex set and geodesic Econvex 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 Econvex 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 Econvex on a geodesic Econvex 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 Econvex sets and geodesic strongly Econvex functions
In this section, we introduce a geodesic strongly Econvex (GSEC) set and a geodesic strongly Econvex (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 2dimensional simply complete Riemannian manifold of nonpositive sectional curvature, and \(B\subset N^{2} \) be an open starshaped. 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 tconvex 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 \(\gamma\in[0,1] \),
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 realvalued 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
If \(\alpha=0 \), then
If \(b_{1}, b_{2}\geq0 \), then
On the other hand
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
On the other hand
It follows that
if and only if
if and only if
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

(2)
Every gconvex 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
□
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 nondecreasing 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 \(\gamma\in[0,1] \),
Since \(f_{2} \) is a nondecreasing convex function,
which means that \(f_{2}\circ f_{1} \) is a GSEC function on B. Similarly, if \(f_{2} \) is a strictly nondecreasing 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
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
It follows that
Then
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 realvalued 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
Then
Hence,
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,
\(\forall b_{1},b_{2}\in B \), \(\alpha\in[0,1]\), 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.
An epigraph of f is given by
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 realvalued function and \(F\colon\mathbb{R}\rightarrow\mathbb{R} \) be a map such that \(F(f(b)+a)=f(E(b))+a \), for each nonnegative 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
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
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
\(\forall\alpha\in[0,1]\) and \(\gamma\in[0,1] \). Therefore,
\(\forall\alpha\in[0,1]\) 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 nonnegative real number a. Suppose that \(\lbrace f_{j}, j\in I \rbrace \) is a family of realvalued 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
are geodesic strongly \(E\times F \)convex on \(B\times\mathbb{R} \). Therefore,
is geodesic strongly \(E\times F \)convex set. Then, according to Theorem 4.2 we see that f is a GSEC function on B. □
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Kılıçman, A., Saleh, W. On geodesic strongly Econvex sets and geodesic strongly Econvex functions. J Inequal Appl 2015, 297 (2015). https://doi.org/10.1186/s136600150824z
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DOI: https://doi.org/10.1186/s136600150824z