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R E S E A R C H

Open Access

On geodesic strongly

E

-convex sets and

geodesic strongly

E

-convex functions

Adem Kılıçman

*

and Wedad Saleh

*Correspondence:

akilic@upm.edu.my Department of Mathematics, University Putra Malaysia, Serdang, Malaysia

Abstract

In this article, geodesicE-convex sets and geodesicE-convex functions on a Riemannian manifold are extended to the so-called geodesic stronglyE-convex sets and geodesic stronglyE-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.

MSC: 52A20; 52A41; 53C20; 53C22

Keywords: geodesicE-convex sets; geodesicE-convex functions; Riemannian manifolds

1 Introduction

Convexity and its generalizations play an important role in optimization theory, convex analysis, Minkowski space, and fractal mathematics [–]. In order to extend the valid-ity of their results to large classes of optimization, these concepts have been general-ized and extended in several directions using novel and innovative techniques. Youness

[] definedE-convex sets andE-convex functions, which have some important

applica-tions in various branches of mathematical sciences [–]. However, some results given by Youness [] seem to be incorrect according to Yang []. Chen [] extendedE-convexity

to a semi-E-convexity and discussed some of there properties. Also, Youness and Emam

[] discussed a new class functions which is called stronglyE-convex functions by tak-ing the images of two pointsxandx under an operatorE:Rn→Rnbesides the two

points themselves. StrongE-convexity was extended to a semi-strongE-convexity as well as quasi- and pseudo-semi-strongE-convexity in []. The authors investigated the char-acterization of efficient solutions for multi-objective programming problems involving semi-strongE-convexity [].

A generalization of convexity on Riemannian manifolds was proposed by Rapcsak [] and Udriste []. Moreover, Iqbalet al.[] introduced geodesicE-convex sets and

geodesicE-convex functions on Riemannian manifolds.

Motivated by earlier research works [, –] and by the importance of the concepts of convexity and generalized convexity, we discuss a new class of sets on Riemannian man-ifolds and a new class of functions defined on them, which are called geodesic strongly

E-convex sets and geodesic stronglyE-convex functions, and some of their properties are presented.

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2 Preliminaries

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

LetNbe aCm-dimensional Riemannian manifold, andTzNbe the tangent space to

Natz. Also, assume thatμz(x,x) is a positive inner product on the tangent spaceTzN

(x,x∈TzN), which is given for each point of N. Then aC∞ mapμ: zμz, which

assigns a positive inner productμz toTzNfor each pointzofNis called a Riemannian

metric.

The length of a piecewiseCcurveη: [a

,a]→Nwhich is defined as follows:

L(η) =

a

a

η´(x)dx.

We define d(z,z) =inf{L(η) : ηis a piecewiseCcurve joiningztoz} for any points z,z∈N. Thendis a distance which induces the original topology onN. As we know on

every Riemannian manifold there is a unique determined Riemannian connection, called a Levi-Civita connection, denoted byXY, for any vector fieldsX,YN. Also, a smooth

pathηis a geodesic if and only if its tangent vector is a parallel vector field along the path

η,i.e.,ηsatisfies the equationη(´t)η´(t) = . Any pathηjoiningzandzinNsuch that L(η) =d(z,z) is a geodesic and is called a minimal geodesic.

Finally, assume that (N,η) is a completem-dimensional Riemannian manifold with Rie-mannian connection. Letx,x∈Nandη: [, ]→Nbe a geodesic joining the points xandx, which means thatηx,x() =xandηx,x() =x.

Definition .[] A setBin a Riemannian manifoldNis called totally convex ifB con-tains every geodesicηx,x ofNwhose endpointsxandxbelong toB.

Note the whole of the manifoldNis 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 [].

Theorem .[] The intersection of any number of a totally convex sets is totally con-vex.

Remark . In general, the union of a totally convex set is not necessarily totally con-vex.

Definition .[] A functionf:B→Ris called a geodesic convex function on a totally convex setBNif for every geodesicηx,x, then

fηx,x(γ)

γf(x) + ( –γ)f(x)

holds for allx,x∈Bandγ∈[, ].

In , stronglyE-convex sets and stronglyE-convex functions were introduced by

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Definition .[]

() A subsetB⊆Rnis called a stronglyE-convex set if there is a mapE:RnRnsuch

that

γαb+E(b)

+ ( –γ)αb+E(b)

B

for eachb,b∈B,α∈[, ]andγ ∈[, ].

() A functionf: B⊆RnRis called a stronglyE-convex function onNif there is a

mapE:RnRnsuch thatBis a stronglyE-convex set and

fγαb+E(b)

+ ( –γ)αb+E(b)

γfE(b)

+ ( –γ)fE(b)

for eachb,b∈B,α∈[, ]andγ ∈[, ].

In , the geodesicE-convex set and geodesicE-convex functions on a Riemannian

manifold were introduced by Iqbalet al.[] as follows.

Definition .[]

() Assume thatE:NNis a map. A subsetBin a Riemannian manifoldNis called geodesicE-convex iff there exists a unique geodesicηE(b),E(b)(γ)of lengthd(b,b), which belongs toB, for eachb,b∈Bandγ ∈[, ].

() A functionf: BN→Ris called geodesicE-convex on a geodesicE-convex setB if

fηE(b),E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

for allb,b∈Bandγ ∈[, ].

3 Geodesic stronglyE-convex sets and geodesic stronglyE-convex functions

In this section, we introduce a geodesic strongly E-convex (GSEC) set and a geodesic

stronglyE-convex (GSEC) function in a Riemannian manifoldNand discuss some of their

properties.

Definition . Assume thatE: NN is a map. A subsetBin a Riemannian manifold

N is called GSEC if and only if there is a unique geodesicηαb+E(b),αb+E(b)(γ) of length d(b,b), which belongs toB,∀b,b∈B,α∈[, ], andγ ∈[, ].

Remark .

() Every GSEC set is a GEC set whenα= .

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

Example . LetNbe a -dimensional simply complete Riemannian manifold of

non-positive sectional curvature, andBNbe an open star-shaped. LetE:NNbe a

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Proposition . Every convex set BN is a GSEC set.

Proof Let us take a mapE: NNsuch asE=IwhereIis the identity map andα= ,

then we have the required result.

Note if we take the mappingE(x) = ( –α)x,xB, then the definition of a GSE reduces to the definition of at-convex set.

Theorem . If BN is a GSEC set,then E(B)⊆B.

Proof SinceBis a GSEC set, we have for eachb,b∈B,α∈[, ], andγ∈[, ],

ηαb+E(b),αb+E(b)(γ)∈B.

Forγ=  andα= , we haveηE(b),E(b)() =E(b)∈B, thenE(B)⊆B.

Theorem . If{Bj,jI}is an arbitrary family of GSEC subsets of N with respect to the

mapping E:NN,then the intersectionjIBjis a GSEC subset of N.

Proof IfjIBjis an empty set, then it is obviously a GSEC subset of N. Assume that

b,b∈

jIBj, thenb,b∈Bj,∀jI. By the GSEC ofBj, we getηαb+E(b),αb+E(b)(γ)∈Bj, ∀jI,α∈[, ], andγ ∈[, ]. Hence,ηαb+E(b),αb+E(b)(γ)∈

jIBj,∀α∈[, ] andγ

[, ].

Remark . The above theorem is not generally true for the union of GSEC subsets ofN.

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

Definition . A real-valued functionf:BN→Ris said to be a GSEC function on a GSEC setB, if

fηαb+E(b),αb+E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

,

b,b∈B,α∈[, ], andγ ∈[, ]. If the above inequality is strict for allb,b∈B,αb+ E(b)=αb+E(b),α∈[, ], andγ ∈(, ), thenf is called a strictly GSEC function.

Remark .

() Every GSEC function is a GEC function whenα= . The following example shows that a GEC function is not necessarily a GSEC function.

Example . Consider the function f: R→R where f(b) = –|b| and suppose that

E:R→Ris given asE(b) = –b. We consider the geodesicηsuch that

ηαb+E(b),αb+E(b)(γ) =

⎧ ⎨ ⎩

–[αb+E(b) +γ(αb+E(b) –αb–E(b))]; bb≥,

–[αb+E(b) +γ(αb+E(b) –αb–E(b))]; bb< 

=

⎧ ⎨ ⎩

–[(α– )b+γ((α– )b+ ( –α)b)]; bb≥,

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Ifα= , then

ηE(b),E(b)(γ) =

⎧ ⎨ ⎩

[b+γ(b–b)]; bb≥,

[b+γ(b–b)]; bb< .

Ifb,b≥, then

fηE(b),E(b)(γ)

=fb+γ(b–b)

= –( –γ)b+γb

.

On the other hand

γfE(b)

+ ( –γ)fE(b)

=γf(–b) + ( –γ)f(–b) = –

( –γ)b+γb

.

Hence,f(ηE(b),E(b)(γ))≤γf(E(b)) + ( –γ)f(E(b)),∀γ ∈[, ].

Similarly, the above inequality holds true whenb,b< .

Now, letb< ,b> , then

fηE(b),E(b)(γ)

=fb+γ(b–b)

= –( +γ)b–γb

.

On the other hand

γfE(b)

+ ( –γ)fE(b)

=γf(–b) + ( –γ)f(–b) =γb– ( –γ)b.

It follows that

fηE(b),E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

if and only if

–( +γ)b–γb

γb– ( –γ)b

if and only if

–γb≤,

which is always true for allγ ∈[, ].

Similarly,f(ηE(b),E(b)(γ))≤γf(E(b)) + ( –γ)f(E(b)),∀γ ∈[, ] also holds forb> 

andb< .

Thus,f is a GEC function onR, but it is not a GSEC function because if we takeb= , b= – andγ=, then

fηαb+E(b),αb+E(b)(γ)

=f

 α

 

= 

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>  f

E()+ f

E(–)

=–

 , ∀α∈(, ].

() Everyg-convex functionf on a convex setBis a GSEC function whenα= andEis the identity map.

Proposition . Assume that f:B→Ris a GSEC function on a GSEC set BN,then f(αb+E(b))≤f(E(b)),∀bB andα∈[, ].

Proof Sincef:B→Ris a GSEC function on a GSEC setBN, thenηαb+E(b),αb+E(b)(γ)∈ B,∀b,b∈B,α∈[, ], andγ ∈[, ]. Also,

fηαb+E(b),αb+E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

thus, forγ = , we getηαb+E(b),αb+E(b)(γ) =αb+E(b). Then

fαb+E(b)

fE(b)

.

Theorem . Consider that BN is a GSEC set and f:B→Ris a GSEC function.If f:I→Ris a non-decreasing convex function such thatrang(f)⊂I,then f◦fis a GSEC function on B.

Proof Sincefis a GSEC function, for allb,b∈B,α∈[, ], andγ ∈[, ],

f

ηαb+E(b),αb+E(b)(γ)

γf

E(b)

+ ( –γ)f

E(b)

.

Sincefis a non-decreasing convex function,

f◦f

ηαb+E(b),αb+E(b)(γ)

=f

f

ηαb+E(b),αb+E(b)(γ)

f

γf

E(b)

+ ( –γ)f

E(b)

γf

f

E(b)

+ ( –γ)f

f

E(b)

=γ(f◦f)

E(b)

+ ( –γ)(f◦f)

E(b)

,

which means thatf◦fis a GSEC function onB. Similarly, iffis a strictly non-decreasing

convex function, thenf◦fis a strictly GSEC function.

Theorem . Assume that BN is a GSEC set and fj:B→R,j= , , . . . ,m are GSEC

functions.Then the function

f =

m

j= njfj

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Proof Sincefj,j= , , . . . ,mare GSEC functions,∀b,b∈B,α∈[, ], andγ ∈[, ], we

have

fj

ηαb+E(b),αb+E(b)(γ)

γfj

E(b)

+ ( –γ)fj

E(b)

.

It follows that

njfj

ηαb+E(b),αb+E(b)(γ)

γnjfj

E(b)

+ ( –γ)njfj

E(b)

. Then m j= njfj

ηαb+E(b),αb+E(b)(γ)

γ

m

j= njfj

E(b)

+ ( –γ)

m

j= njfj

E(b)

=γfE(b)

+ ( –γ)fE(b)

.

Thus,f is a GSEC function.

Theorem . Let BN be a GSEC set and{fj,jI}be a family of real-valued functions

defined on B such thatsupjIfj(b)exists inR,∀bB.If fj: B→R,jI are GSEC functions

on B,then the function f:B→R,defined by f(b) =supjIfj(b),∀bB is GSEC on B.

Proof Sincefj,jIare GSEC functions on a GSEC setB,∀b,b∈B,α∈[, ], andγ

[, ], we have

fj

ηαb+E(b),αb+E(b)(γ)

γfj

E(b)

+ ( –γ)fj

E(b)

.

Then

sup jI

fj

ηαb+E(b),αb+E(b)(γ)

≤sup jI

γfj

E(b)

+ ( –γ)fj

E(b)

=γsup jI

fj

E(b)

+ ( –γ)sup jI

fj

E(b)

=γfE(b)

+ ( –γ)fE(b)

.

Hence,

fηαb+E(b),αb+E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

,

which means thatf is a GSEC function onB.

Proposition . Assume that hj:N→R,j= , , . . . ,m are GSEC functions on N,with

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Proof Sincehj,j= , , . . .mare GSEC functions,

hj

ηαb+E(b),αb+E(b)(γ)

γhj

E(b)

+ ( –γ)hj

E(b)

≤,

b,b∈B,α∈[, ], andγ ∈[, ]. SinceE(B)⊆B,ηαb+E(b),αb+E(b)(γ)∈B. Hence,Bis

a GSEC set.

4 Epigraphs

Youness and Emam [] defined a strongly E×F-convex set whereE: RnRn and

F: R→Rand studied some of its properties. In this section, we generalize a strongly

E×F-convex set to a geodesic stronglyE×F-convex set on Riemannian manifolds and

discuss GSEC functions in terms of their epigraphs. Furthermore, some properties of GSE sets are given.

Definition . LetBN×R, E: NN andF: R→R. A set Bis called geodesic stronglyE×F-convex if (b,β), (b,β)∈Bimplies

ηαb+E(b),αb+E(b)(γ),γF(β) + ( –γ)F(β)

B

for allα∈[, ] andγ∈[, ].

It is not difficult to prove thatBN is a GSEC set if and only ifB×Ris a geodesic stronglyE×F-convex set.

An epigraph off is given by

epi(f) =(b,a) :bB,a∈R,f(b)≤a.

A characterization of a GSEC function in terms of itsepi(f) is given by the following the-orem.

Theorem . Let E:NN be a map,BN be a GSEC set,f:B→Rbe a real-valued function and F:R→Rbe 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 ifepi(f)is geodesic strongly E×F-convex on B×R.

Proof Assume that (b,a), (b,a)∈epi(f). IfBis a GSEC set, thenηαb+E(b),αb+E(b)(γ)∈ B, ∀α∈[, ] andγ ∈[, ]. Since E(b)∈Bforα= , γ = , alsoE(b)∈Bforα= ,

γ = , let F(a) and F(a) be such that f(E(b))≤F(a) and f(E(b))≤F(a). Then

(E(b),F(a)), (E(b),F(a))∈epi(f).

Letf be GSEC onB, then

fηαb+E(b),αb+E(b)(γ)

γfE(b)

+ ( –γ)fE(b)

γF(a) + ( –γ)F(a).

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Conversely, assume thatepi(f) is geodesic stronglyE×F-convex onB×R. Letb,b∈B,

α∈[, ], andγ ∈[, ], then (b,f(b))∈epi(f) and (b,f(b))∈epi(f). Now, sinceepi(f)

is geodesic stronglyE×F-convex onB×R, we obtain (ηαb+E(b),αb+E(b)(γ),γF(f(b)) +

( –γ)F(f(b)))∈epi(f), then

fηαb+E(b),αb+E(b)(γ)

γFf(b)

+ ( –γ)Ff(b)

=γfE(b)

+ ( –γ)fE(b)

.

This shows thatf is a GSEC function onB.

Theorem . Assume that{Bj,jI}is a family of geodesic strongly E×F-convex sets.

Then the intersectionjIBjis a geodesic strongly E×F-convex set.

Proof Assume that (b,a), (b,a)∈

jIBj, so∀jI, (b,a), (b,a)∈Bj. SinceBjis the

geodesic stronglyE×F-convex sets∀jI, we have

ηαb+E(b),αb+E(b)(γ),γF(a) + ( –γ)F(a)

Bj,

α∈[, ] andγ ∈[, ]. Therefore,

ηαb+E(b),αb+E(b)(γ),γF(a) + ( –γ)F(a)

jI

Bj,

α∈[, ] andγ ∈[, ]. ThenjIBjis a geodesic stronglyE×F-convex set.

Theorem . Assume that E:NN and F:R→Rare two maps such that F(f(b) +a) =

f(E(b)) +a for each non-negative real number a.Suppose that{fj,jI}is a family of

real-valued functions defined on a GSEC set BN which are bounded from above.Ifepi(fj)are

geodesic strongly E×F-convex sets,then the function f which is given by f(b) =supjIfj(b), ∀bB,is a GSEC function on B.

Proof If eachfj,jIis a GSEC function on a GSEC geodesic setB, then

epi(fj) =

(b,a) :bB,a∈R,fj(b)≤a

are geodesic stronglyE×F-convex onB×R. Therefore,

jI

epi(fj) =

(b,a) :bB,a∈R,fj(b)≤a,jI

=(b,a) :bB,a∈R,f(b)≤a

is geodesic stronglyE×F-convex set. Then, according to Theorem . we see thatf is a

GSEC function onB.

Competing interests

(10)

Authors’ contributions

All authors jointly worked on deriving the results and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.

Received: 1 May 2015 Accepted: 15 September 2015

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