Open Access
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

© Wuhan University 2022

Licence Creative CommonsThis 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.[1-7]), has attracted much attention in various mathematics areas, such as analysis, geometry and optimization theory etc. (Refs.[8-14]). 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.[15-17] 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 so-called 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 n-space 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 1-dimensional subspace generated by a nonzero , we write and simply instead of and , respectively, where denotes the orthogonal complementary space of . is a 0-sphere and is an -sphere for each nonzero . In geometric language, and in are called (a pair of) antipodes. So a 0-sphere consists of a pair of antipodes. For the s-convexity 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 so-called spherically convex combination (s-convex combination for brevity) of and non-negative 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 (s-convex for brevity) if

whenever , with .

If further contains no antipodes, then is called a proper s-convex set.

Remark 1   It is easy to check that all -spheres are s-convex, and it was shown in Ref.[17] that if is s-convex, 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 s-convex 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 s-convex set. A function is called spherically convex (s-convex for brevity) if

holds for , with .

For an s-convex 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 s-ball. A set is called s-open if for each , there is such that . It is easy to see that is s-open 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 s-convex 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 s-directional derivative and s-Gateaux differentiability.

Definition 3   Let and be a function. If for some , the limit

exists, where , then is called the s-directional 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 s-Gateaux differentiable (s-G-differentiable 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 s-directional derivative . Also, in a same manner, one may construct a function such that (resp.) exists, but (resp.) does not. Therefore, the s-directional 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 s-gradient of functions defined on .

Definition 4   Let and be a function s-G-differentiable at . If satisfies ,then is called an s-subgradient of at . If the s-subgradient is unique, then it is called the s-gradient of at , denoted by .

Examples are easily found to show that, for a function defined in an open neighborhood of , its s-gradient 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 s-gradient 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 s-open set containing . If there is , such that

whenever and , then is called s-Frechet differentiable (s-F-differentiable for brevity or s-differentiable simply) at , and is called the s-F-derivative of at.

Remark 3   It is easy to check that if is differentiable at , then is s-differentiable 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 s-differentiable but not differentiable at . Also, it is easy to check that the s-differentiability implies the s-G-differentiability (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 s-open set containing and s-differentiable at , then and

Proof   Since

whenever and , we have

from which the conclusions follow.

The following proposition provides the expressions of s-gradients for s-differentiable 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 s-open containing and s-differentiable at , then for arbitrary pairly orthogonal (that is, form a standard orthogonal basis of ), we have

Proof   Since is s-differentiable at , it is easy to check that exists and for each , in particular, . So,

which implies clearly .

3 Criterions of s-Convex Functions

In this section, we discuss the criterions of s-convexity for functions defined on in terms of s-gradients. The results obtained here are improvement of several criterions of s-convexity 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 s-open s-convex set and be s-differentiable. Then is s-convex on if and only if, for any and ,

holds for any .

Proof   () Suppose is s-convex on . By the s-differentiability of , for any and , we have

whenever .

Denote for and . Then clearly, so by the s-convexity of , we have

which leads to

that is

(1)

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 s-open (one may find the argument easily). Thus, for any with , i.e. and , we have

since and . Therefore

(2)

With the same arguments on the pairs , and , respectively, we have as well

(3)

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 s-convexity, we introduce the following concept.

Definition 6   Let be an s-open s-convex set and be a map. Then, is called s-monotone on if for any with ,

holds, where , such that

In terms of the s-monotonicity we have the following improvement of Proposition 8 in Ref.[15].

Theorem 3   Let be an s-open s-convex set and be s-differentiable. Then is s-convex if and only if is s-monotone on .

Proof   () Suppose is s-convex on . Then for any with and , with , , we have by Theorem 2

which leads clearly to

and in turn

since when , i.e. is s-monotone on .

() Conversely, suppose is s-monotone on. If contains antipodes, then, as explained in the proof of Theorem 2, . Thus, for any and , we have

(4)

which leads to . Hence, since and is arbitrary, we obtain

Thus, by the s-differentiability of , for , we have

(5)

(6)

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 s-convex 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 middle-point convex (i.e.) and in turn convex since it is clearly continuous on . Thus, for any , we have

By the arbitrariness of and , is s-convex.

Final Remark In this paper, we introduce first the concepts of s-directional derivative, s-gradient and s-Gateaux differentiability and s-Frechet 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 s-convexity 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.

References

  1. Robinson C V. Spherical theorems of Helly type and congruence indices of spherical caps[J]. Amer J Math, 1942, 64: 260-272. [CrossRef] [MathSciNet] [Google Scholar]
  2. Horn A. Some generalizations of Helly's theorem on convex sets[J]. Bull Amer Math Soc, 1949, 55: 923-929. [CrossRef] [MathSciNet] [Google Scholar]
  3. Katchalski M. A Helly type theorem on the sphere[J]. Proc of Amer Math Soc, 1977, 66(1) :119-122. [CrossRef] [MathSciNet] [Google Scholar]
  4. Baker M J C. A spherical Helly-type theorem[J]. Pacific J of Math, 1967, 23(1): 1-3. [CrossRef] [MathSciNet] [Google Scholar]
  5. Danzer L, Grünbaum B, Klee V. Helly's theorem and its relatives convexity[J]. Proc of Symp in Pure Math, 1963, 7: 99-180. [CrossRef] [Google Scholar]
  6. Cattle R W, Randow R, Stone R E. On spherically convex sets and Q-matrices[J]. Lin Alg and Its Appl, 1981, 41: 73-80. [CrossRef] [Google Scholar]
  7. Drezner Z, Wesolowsky G O. Minimax and maximin facility location problems on a sphere[J]. Naval Res Logist Quart, 1983, 30: 305-312. [CrossRef] [MathSciNet] [Google Scholar]
  8. Besau F, Schuster F E. Binary operations in spherical convex geometry[J]. India Univ Math J, 2016, 65(4): 1263-1288 [CrossRef] [Google Scholar]
  9. Das P, Chakraborti N R, Chaudhuri P K. Spherical minimax location problem[J]. Comput Optim Appl, 2001, 18: 311-326. [CrossRef] [MathSciNet] [Google Scholar]
  10. Lassak M. Width of spherical convex bodies[J]. Aequationes Math, 2015, 89: 555-567. [CrossRef] [MathSciNet] [Google Scholar]
  11. Ferreira O P, Iusem A N, Nemeth S Z. Projections on to convex sets on the sphere[J]. J Glob Optim, 2013, 57: 663-676. [CrossRef] [Google Scholar]
  12. Ferreira O P, Nemeth S Z. On the spherical convexity of quadratic functions[J]. J Glob Optim, 2019, 73: 537-545. [CrossRef] [Google Scholar]
  13. Gao F, Hug D, Schneider R. Intrinsic volumes and polar sets in spherical space[J]. Math Notae, 2003, 41:159-176. [Google Scholar]
  14. Zhou X, Guo Q. Compositions and transformations on compositions and transformations on spherical convex sets[J]. Wuhan Univ J of Nat Sci, 2020, 25(4): 277-285. [Google Scholar]
  15. Ferreira O P, Iusem A N, Nemeth S Z. Concepts and techniques of optimization on the sphere[J]. TOP, 2014 , 22(3):1148-1170. [CrossRef] [MathSciNet] [Google Scholar]
  16. Guo Q, Peng Y L. Spherically convex sets and spherically convex functions[J]. J of Conv Anal, 2021, 28(1): 103-122. [Google Scholar]
  17. Guo Q. Convexity theory on spherical spaces(I) [J]. Sci China: Math(A), 2020, 50(12): 1745- 1772(Ch). [Google Scholar]
  18. Shao Y C, Guo Q. An analytic approach to spherically convex sets in Formula [J]. J of Math (China), 2018, 38 : 473-480(Ch). [Google Scholar]
  19. Zhang Y, Guo Q. Properties and criterions of spherically convex functions[J]. J Suzhou Univ of Sci and Tech (Nat Sci), 2022, 39(1): 21-26+46(Ch). [Google Scholar]

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