Issue 
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 4, August 2022



Page(s)  273  280  
DOI  https://doi.org/10.1051/wujns/2022274273  
Published online  26 September 2022 
Mathematics
CLC number: O 18
Differentiability of Functions on Spheres and Criterions of Convexity
Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{†} To whom correspondence should be addressed. Email: guoqi@mail.usts.edu.cn
Received:
10
March
2022
Some basic concepts for functions defined on subsets of the unit sphere, such as the sdirectional derivative, sgradient and sGateaux and sFrechet differentiability etc, are introduced and investigated. These concepts are different from the usual ones for functions defined on subsets of Euclidean spaces, however, the results obtained here are very similar. Then, as applications, we provide some criterions of sconvexity for functions defined on unit spheres which are improvements or refinements of some known results.
Key words: spherical convexity / spherically convex function / Gateaux and Frechet differentiability / criterion of convexity
Biography: CHEN Tingting, female, Master candidate, research direction: convex geometry. Email: 1970209203@qq.com
Fundation item: Supported by the National Natural Science Foundation of China (12071334, 11671293)
© Wuhan University 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
In the past two decades, the convexity theory on spherical spaces, emerging almost at the same time as that on Euclidean (linear) spaces and developing relatively slow in the last century (Refs.[17]), has attracted much attention in various mathematics areas, such as analysis, geometry and optimization theory etc. (Refs.[814]). Encouragingly, some efforts have been made to establish a systematic theory, parallel to that on Euclidean spaces, of convexity on spherical spaces (for details see Refs.[1517] and the references therein).
Although some progresses have been made in the study on spherical convexity, the task to establish a systematic theory is far away from being completed, simply because many counterparts of concepts and definitions for convex sets in Euclidean spaces have not been found for spherical convexity, due to the lack of suitable compositions and operators on spheres. So, in this paper, we make an effort to study the socalled spherical differentiability of functions defined on spheres and the criterions of spherically convex functions.
The paper is organized as follows. In Section 1, we recall some notations, definitions and basic properties about spherically convex sets and spherically convex functions which will be used throughout the paper. In Section 2, we introduce the spherical Gateaux and spherical Frechet differentiability of functions defined merely on spheres. These differentiabilities are proper extensions of those defined in Ref.[15] where the authors have to assume that the functions are defined on some suitable Euclidean open sets. Section 3 is devoted to studying the criterions of spherical convexity of functions defined on spheres. The results obtained here generalize those in Ref.[15] etc. and will play some roles in the further study.
1 Preliminaries
, denote the Euclidean nspace and the unit sphere in , respectively. As usual,,denote the standard inner product and the norm induced by on , respectively. Often we also view as an affine space, so we will not distinguish vectors and points intentionally. The origin (or zero vector) of is always denoted by the letter .
A set of the form , where is a dimensional subspace of , is called a sphere. If is a 1dimensional subspace generated by a nonzero , we write and simply instead of and , respectively, where denotes the orthogonal complementary space of . is a 0sphere and is an sphere for each nonzero . In geometric language, and in are called (a pair of) antipodes. So a 0sphere consists of a pair of antipodes. For the sconvexity of sets or functions on , there are several equivalent definitions, among which the one given in Refs.[16, 17] is the only analytic form. So, we follow the approach adopted in Refs.[16, 17].
The spherical addition, denoted by "", in (see Refs.[16, 18]) is defined by , , where , called the radial projection, is defined by
which is clear of the properties:
i) ;
ii) for and ;
iii) if and only if or.
The spherical addition is communicative but not associative, so the following composition is introduced in Refs.[16, 18]: for , define
Naturally, when , we write instead of . In terms of the spherical addition, the socalled spherically convex combination (sconvex combination for brevity) of and nonnegative with is defined as
Now, we introduce the definition of spherically convex sets given in Ref.[16] (for sets containing no antipodes) and in Ref.[17] (for general cases).
Definition 1 A subset is called spherically convex (sconvex for brevity) if
whenever , with .
If further contains no antipodes, then is called a proper sconvex set.
Remark 1 It is easy to check that all spheres are sconvex, and it was shown in Ref.[17] that if is sconvex, then is contained in some closed hemisphere.
Denote, for ,
which is a subset of . When , is called the short arc connecting and (it can be checked easily that the short arc defined here coincides with the usual one defined in geometric languages), and for . For , , and are defined in a similar manner. From these notations, we see that the definition here is equivalent to the popular one adopted by other authors recently (see Refs.[16, 17] for the precise proof).
For and , denote
which are semicircles of various types passing through .
Next, we recall the concept of spherically convex functions. In fact, before the "spherical convex combination" composition is introduced, it was not as simple as one may think to define spherically convex functions. Ferreira et al in Ref.[15] proposed a definition in terms of minimal geodesic segment (function): if is an sconvex set and is a function, then is called spherically convex if the function is a univariate convex function for each minimal geodesic segment (function) (see Ref.[15] for more information). This definition is quite intuitive but not very convenient in application. Here, we adopt the one given in Refs.[16, 17] which is equivalent to Ferreira's in Ref.[15].
Definition 2 Let be an sconvex set. A function is called spherically convex (sconvex for brevity) if
holds for , with .
For an sconvex function on , it was shown in Ref.[19] that it is continuous with respect to the both metrics mentioned below, and also that restricted on each sphere contained in , it is a constant, in particular, if .
The intrinsic metric in , defined by for , will be used in studying the continuity and the differentiability of functions on . An elementary geometric argument shows, so the intrinsic metric and Euclidean metric are equivalent on .
For given and , the set
is called an sball. A set is called sopen if for each , there is such that . It is easy to see that is sopen if and only if there is an open set such that .
For with , to describe the points in , there are two popular geodesic segment functions connecting considered in Refs.[16, 17, 19] and Ref.[15], respectively:
where . It can be easily checked that (see Ref.[15] for more information).
The following conclusion, cited as a lemma here (a proof is also included for convenience), was confirmed in Ref.[19].
Lemma 1 Let with . Then if and only if
(or equivalently, ).
Consequently,
for all .
Proof Clearly, iff . For brevity, denote . By an elementary geometric argument, we see that if and only if (noticing),
and in turn if and only if
Finally, another elementary geometric argument (drawing a picture to check) shows that
Therefore,
2 Differentiability of Functions on Sphere
Ferreira et al in Ref.[15] defined and studied the spherical Frechet differentiability only for the functions differentiable on an open set containing an sconvex set. More precisely, they simply took the orthogonal projection of gradients of the functions (to a suitable subspace) as the spherical gradients. Clearly, such an approach limits the applications of their results. In this section, we will define and study the Gateaux differentiability and Frechet differentiability of functions defined merely on a subset of .
First, we give the concepts of sdirectional derivative and sGateaux differentiability.
Definition 3 Let and be a function. If for some , the limit
exists, where , then is called the sdirectional derivative of at along (direction) .
If both and exist and , then is called the partial derivative of at along (direction) , denoted by .
If exists for all , then is called sGateaux differentiable (sGdifferentiable for brevity) at .
Remark 2 i) in Definition 1 can be replaced by since .
ii) Naturally, for each nonzero , one may define as well (if exists), where . However, we point out that such an extended definition is not essentially necessary: writing for some and , we have by the property of ,
where , and in turn iff .
Thus, exists iff exists, and in such a case .
Consider the function defined by or 0 (otherwise), where is fixed. It is easy to check that, for each , the (usual) directional derivative
while the sdirectional derivative . Also, in a same manner, one may construct a function such that (resp.) exists, but (resp.) does not. Therefore, the sdirectional derivative and the (usual) directional derivative (along direction ) have nothing to do with each other in general. However, we have the following conclusion.
Proposition 1 Let and be an open set containing . If a function is differentiable at , then for each , exists and
Proof We have for since is differentiable at , where denotes the usual gradient of .
Thus, to calculate for , we compute first. Denoting (observing that iff ), we have
and in turn
Therefore, we obtain
where we used the fact since . The proof completes.
Next, we define the sgradient of functions defined on .
Definition 4 Let and be a function sGdifferentiable at . If satisfies ,then is called an ssubgradient of at . If the ssubgradient is unique, then it is called the sgradient of at , denoted by .
Examples are easily found to show that, for a function defined in an open neighborhood of , its sgradient and its usual gradient have nothing to do with each other in general. However, we have the following proposition.
Proposition 2 Let and be an open neighborhood of . If a function is differentiable at , then exists and
where denotes the orthogonal projection from to . In turn, for .
Proof Writing
we have, as shown in the proof of Proposition 1, for each ,
since , which implies clearly exists and , and in turn
for all .
Ferreira et al in Ref.[15] took as the definition of sgradient directly. Obviously, such a definition may make sense only for functions satisfying the conditions as in Proposition 2. To illustrate our definition as a useful proper extension of Ferreira's, we introduce the following concept.
Definition 5 Let and be a function, where is an sopen set containing . If there is , such that
whenever and , then is called sFrechet differentiable (sFdifferentiable for brevity or sdifferentiable simply) at , and is called the sFderivative of at.
Remark 3 It is easy to check that if is differentiable at , then is sdifferentiable at . However, the inverse is not true even if the function is defined in an open neighborhood of , as shown by the following example: consider the function defined by
() or 1 (otherwise),
where is fixed. It is easy to see that is sdifferentiable but not differentiable at . Also, it is easy to check that the sdifferentiability implies the sGdifferentiability (however, the inverse is not true). More precisely, we have the following improvement of Proposition 2.
Theorem 1 If a function is defined on an sopen set containing and sdifferentiable at , then and
Proof Since
whenever and , we have
from which the conclusions follow.
The following proposition provides the expressions of sgradients for sdifferentiable functions, which has its own significance clearly even if it will not be used in this paper.
Proposition 3 If a function is defined on an sopen containing and sdifferentiable at , then for arbitrary pairly orthogonal (that is, form a standard orthogonal basis of ), we have
Proof Since is sdifferentiable at , it is easy to check that exists and for each , in particular, . So,
which implies clearly .
3 Criterions of sConvex Functions
In this section, we discuss the criterions of sconvexity for functions defined on in terms of sgradients. The results obtained here are improvement of several criterions of sconvexity in Ref.[15] since the functions considered there have to be differentiable on some open set.
The first one is an improvement of Proposition 7 in Ref.[15].
Theorem 2 Let be an sopen sconvex set and be sdifferentiable. Then is sconvex on if and only if, for any and ,
holds for any .
Proof () Suppose is sconvex on . By the sdifferentiability of , for any and , we have
whenever .
Denote for and . Then clearly, so by the sconvexity of , we have
which leads to
that is
Since by Lemma 1, we have
;
Thus, letting in (1), we obtain
() Suppose for any and ,
hold for all .
If contains antipodes, then since is sopen (one may find the argument easily). Thus, for any with , i.e. and , we have
since and . Therefore
With the same arguments on the pairs , and , respectively, we have as well
Clearly,(2) and (3) lead to . By the arbitrariness of and , is a constant on (noticing that any two circles have intersection) and so is convex.
If contains no antipodes, for any distinct and (nothing to prove for the case ), denote , we have
where (unique) such that (naturally, ). Thus
since
by Lemma 1 and so
To give another criterion of sconvexity, we introduce the following concept.
Definition 6 Let be an sopen sconvex set and be a map. Then, is called smonotone on if for any with ,
holds, where , such that
In terms of the smonotonicity we have the following improvement of Proposition 8 in Ref.[15].
Theorem 3 Let be an sopen sconvex set and be sdifferentiable. Then is sconvex if and only if is smonotone on .
Proof () Suppose is sconvex on . Then for any with and , with , , we have by Theorem 2
which leads clearly to
and in turn
since when , i.e. is smonotone on .
() Conversely, suppose is smonotone on. If contains antipodes, then, as explained in the proof of Theorem 2, . Thus, for any and , we have
which leads to . Hence, since and is arbitrary, we obtain
Thus, by the sdifferentiability of , for , we have
where we used the fact that and . Now, (6) subtracted from (5) gives
which implies (letting ), and in turn, for any ,
Dividing both sides by and then letting , we have
By the arbitrariness of , we have . Hence on by the arbitrariness of . Now, for any distinct with , denoting where , , we have
Therefore is a constant on , in particular, . By the arbitrariness of and , is a constant and in turn sconvex on .
If contains no antipodes, then for any distinct and , denote , , for brevity first.
Then, define , , and ,. Thus, by Lemma 1,
Therefore,
Let (unique) such that ,then clearly,
where denotes the orthogonal projection from to . Observing which implies
we obtain
Now, for any (observing
in such a case), choosing such that and such that , we have, by the monotonicity of ,
where simply because when .
So, is increasing and in turn is convex on , in particular,
which, together with the arbitrariness of and , implies clearly that is middlepoint convex (i.e.) and in turn convex since it is clearly continuous on . Thus, for any , we have
By the arbitrariness of and , is sconvex.
Final Remark In this paper, we introduce first the concepts of sdirectional derivative, sgradient and sGateaux differentiability and sFrechet differentiability of functions defined on subsets of the unit sphere, which are different from the usual ones for functions defined on subsets of Euclidean spaces. Then, as applications, we provide some criterions of sconvexity for functions on spheres which are improvements of results in Ref.[15]. Also, compared with the work in Ref.[15], our work has some advantages in applications since the functions considered here are defined merely on subsets of spheres.
Acknowledgments
The authors thank the referees very much for their carefully reading the first version of the manuscript and pointing out typos and mistakes.
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