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 pointsxandx 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.
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 aC∞m-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: z→μz, 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 piecewiseCcurve joiningztoz} 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,Y∈N. 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 pathηjoiningzandzinNsuch 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 xandx, which means thatηx,x() =xandηx,x() =x.
Definition .[] A setBin a Riemannian manifoldNis called totally convex ifB con-tains every geodesicηx,x ofNwhose endpointsxandxbelong 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 setB⊂Nif 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
Definition .[]
() A subsetB⊆Rnis called a stronglyE-convex set if there is a mapE:Rn→Rnsuch
that
γαb+E(b)
+ ( –γ)αb+E(b)
∈B
for eachb,b∈B,α∈[, ]andγ ∈[, ].
() A functionf: B⊆Rn→Ris called a stronglyE-convex function onNif there is a
mapE:Rn→Rnsuch 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:N→Nis 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: B⊆N→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: N→N 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, andB⊂Nbe an open star-shaped. LetE:N→Nbe a
Proposition . Every convex set B⊂N is a GSEC set.
Proof Let us take a mapE: N→Nsuch asE=IwhereIis the identity map andα= ,
then we have the required result.
Note if we take the mappingE(x) = ( –α)x,x∈B, then the definition of a GSE reduces to the definition of at-convex set.
Theorem . If B⊂N 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,j∈I}is an arbitrary family of GSEC subsets of N with respect to the
mapping E:N→N,then the intersectionj∈IBjis a GSEC subset of N.
Proof Ifj∈IBjis an empty set, then it is obviously a GSEC subset of N. Assume that
b,b∈
j∈IBj, thenb,b∈Bj,∀j∈I. By the GSEC ofBj, we getηαb+E(b),αb+E(b)(γ)∈Bj, ∀j∈I,α∈[, ], andγ ∈[, ]. Hence,ηαb+E(b),αb+E(b)(γ)∈
j∈IBj,∀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:B⊂N→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≥,
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–
=
> 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 B⊆N,then f(αb+E(b))≤f(E(b)),∀b∈B andα∈[, ].
Proof Sincef:B→Ris a GSEC function on a GSEC setB⊆N, 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 B⊆N 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 Sincefis a GSEC function, for allb,b∈B,α∈[, ], andγ ∈[, ],
f
ηαb+E(b),αb+E(b)(γ)
≤γf
E(b)
+ ( –γ)f
E(b)
.
Sincefis 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◦fis a GSEC function onB. Similarly, iffis a strictly non-decreasing
convex function, thenf◦fis a strictly GSEC function.
Theorem . Assume that B⊆N is a GSEC set and fj:B→R,j= , , . . . ,m are GSEC
functions.Then the function
f =
m
j= njfj
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 B⊆N be a GSEC set and{fj,j∈I}be a family of real-valued functions
defined on B such thatsupj∈Ifj(b)exists inR,∀b∈B.If fj: B→R,j∈I are GSEC functions
on B,then the function f:B→R,defined by f(b) =supj∈Ifj(b),∀b∈B is GSEC on B.
Proof Sincefj,j∈Iare 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 j∈I
fj
ηαb+E(b),αb+E(b)(γ)
≤sup j∈I
γfj
E(b)
+ ( –γ)fj
E(b)
=γsup j∈I
fj
E(b)
+ ( –γ)sup j∈I
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
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: Rn→Rn 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 . LetB⊂N×R, E: N→N 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 thatB⊆N 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) :b∈B,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:N→N be a map,B⊆N 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).
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,j∈I}is a family of geodesic strongly E×F-convex sets.
Then the intersectionj∈IBjis a geodesic strongly E×F-convex set.
Proof Assume that (b,a), (b,a)∈
j∈IBj, so∀j∈I, (b,a), (b,a)∈Bj. SinceBjis the
geodesic stronglyE×F-convex sets∀j∈I, 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)
∈
j∈I
Bj,
∀α∈[, ] andγ ∈[, ]. Thenj∈IBjis a geodesic stronglyE×F-convex set.
Theorem . Assume that E:N→N 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,j∈I}is a family of
real-valued functions defined on a GSEC set B⊆N which are bounded from above.Ifepi(fj)are
geodesic strongly E×F-convex sets,then the function f which is given by f(b) =supj∈Ifj(b), ∀b∈B,is a GSEC function on B.
Proof If eachfj,j∈Iis a GSEC function on a GSEC geodesic setB, then
epi(fj) =
(b,a) :b∈B,a∈R,fj(b)≤a
are geodesic stronglyE×F-convex onB×R. Therefore,
j∈I
epi(fj) =
(b,a) :b∈B,a∈R,fj(b)≤a,j∈I
=(b,a) :b∈B,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
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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