© Wuhan University 2022

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

${\mathbf{R}}^n$, ${\mathbf{S}}^{n-\mathrm{1}}$ denote the Euclidean n-space and the unit sphere in ${\mathbf{R}}^n$, respectively. As usual,$<\cdot ,\cdot >$,$\Vert \cdot \Vert$denote the standard inner product and the norm induced by$<\cdot ,\cdot >$ on ${\mathbf{R}}^n$, respectively. Often we also view ${\mathbf{R}}^n$ as an affine space, so we will not distinguish vectors and points intentionally. The origin (or zero vector) of ${\mathbf{R}}^n$ is always denoted by the letter $o$.

A set of the form ${\mathbf{S}}_V:=V\bigcap {\mathbf{S}}^{n-\mathrm{1}}$, where $V$ is a $\left(k+\mathrm{1}\right)$-dimensional subspace of ${\mathbf{R}}^n$$\left(\mathrm{0}\le k\le n-\mathrm{1}\right)$, is called a $k$-sphere. If $V=<x>$ is a 1-dimensional subspace generated by a nonzero $x\in {\mathbf{R}}^n$, we write ${\mathbf{S}}_x$ and ${\mathbf{S}}_{x^{\perp }}$ simply instead of ${\mathbf{S}}_{<x>}$ and ${\mathbf{S}}_{<x{>}^{\perp }}$, respectively, where $<x{>}^{\perp }$ denotes the orthogonal complementary space of $<x>$. ${\mathbf{S}}_x$ is a 0-sphere and ${\mathbf{S}}_{x^{\perp }}$ is an $\left(n-\mathrm{2}\right)$-sphere for each nonzero $x$. In geometric language, $u$ and $-u$ in ${\mathbf{S}}^{n-\mathrm{1}}$ 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 ${\mathbf{S}}^{n-\mathrm{1}}$, 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 "${+}_s$", in ${\mathbf{R}}^n$ (see Refs.[16, 18]) is defined by $x{+}_{s}y:=\rho (x+y)$, $x,y\in {\mathbf{R}}^n$, where $\rho :{\mathbf{R}}^n\to {\mathbf{S}}^{n-\mathrm{1}}\bigcup \{o\}$, called the radial projection, is defined by

$\rho \left(x\right):=\{\begin{array}{l}\frac{x}{\Vert x\Vert },x\ne o\\ o,x=o\end{array}$

which is clear of the properties:

i) $\rho \circ \rho =\rho$;

ii) $\rho \left(tx\right)=\rho \left(x\right),\rho \left(-x\right)=-\rho \left(x\right)$ for $x\in {\mathbf{R}}^n$and $t>\mathrm{0}$;

iii) $\rho \left(x\right)=x$ if and only if $x\in {\mathbf{S}}^{n-\mathrm{1}}$ or$x=o$.

The spherical addition is communicative but not associative, so the following composition is introduced in Refs.[16, 18]: for $x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_k\in {\mathbf{R}}^n$$\left(k\ge \mathrm{2}\right)$, define

$(s){\displaystyle \sum _{i=\mathrm{1}}^k}{x}_i :=\rho \left({\displaystyle \sum _{i=\mathrm{1}}^k}{x}_i\right)$

Naturally, when $k=\mathrm{2}$, we write $x{+}_{s}y$ instead of $(s)(x+y)$. In terms of the spherical addition, the so-called spherically convex combination (s-convex combination for brevity) of $x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_k\in {\mathbf{R}}^n$ and non-negative ${\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\cdots ,{\lambda }_k$ with ${\displaystyle \sum _{i=\mathrm{1}}^k}{\lambda }_i=\mathrm{1}$ is defined as

$(s){\displaystyle \sum _{i=\mathrm{1}}^k}{\lambda }_i{x}_i (=\rho \left({\displaystyle \sum _{i=\mathrm{1}}^k}{\lambda }_i{x}_i\right)).$

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 $C\subset {\mathbf{S}}^{n-\mathrm{1}}$ is called spherically convex (s-convex for brevity) if

$\lambda u_{\mathrm{1}}{+}_s\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}}\in C$

whenever $u_{\mathrm{1}},u_{\mathrm{2}}\in C$, $\lambda \in [\mathrm{0,1}]$ with $\lambda u_{\mathrm{1}}{+}_s\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}}\ne \mathrm{0}$.

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

Remark 1   It is easy to check that all $k$-spheres are s-convex$\left(\mathrm{0}\le k\le n-\mathrm{1}\right)$, and it was shown in Ref.[17] that if $C\ne {\mathbf{S}}^{n-\mathrm{1}}$ is s-convex, then $C$ is contained in some closed hemisphere.

Denote, for $u_{\mathrm{1}},u_{\mathrm{2}}\in {\mathbf{S}}^{n-\mathrm{1}}$ ,

$[u_{\mathrm{1}},u_{\mathrm{2}}{]}_s:=\{\lambda u_{\mathrm{1}}{+}_s\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}}|\mathrm{0}\le \lambda \le \mathrm{1}\}$

which is a subset of ${\mathbf{S}}^{n-\mathrm{1}}\bigcup \{o\}$. When $u_{\mathrm{1}}\ne u_{\mathrm{2}}$, $[u_{\mathrm{1}},u_{\mathrm{2}}{]}_s$ is called the short arc connecting $u_{\mathrm{1}}$ and $u_{\mathrm{2}}$ (it can be checked easily that the short arc defined here coincides with the usual one defined in geometric languages), and ${[u,-u]}_s=\{o,u,-u\}$ for $u\in {\mathbf{S}}^{n-\mathrm{1}}$. For $u_{\mathrm{1}}\ne u_{\mathrm{2}}$, $(u_{\mathrm{1}},u_{\mathrm{2}}{)}_s$,$(u_{\mathrm{1}},u_{\mathrm{2}}{]}_s$ and $[u_{\mathrm{1}},u_{\mathrm{2}}{)}_s$ 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 $u\in {\mathbf{S}}^{n-\mathrm{1}}$ and $v\in {\mathbf{S}}_u^{\perp }$, denote

${[u,-u)}_s(v):={[u,v]}_s\bigcup {[v,-u)}_s$

${(u,-u)}_s(v):={(u,v]}_s\bigcup {[v,-u)}_s$

${[u,-u]}_s(v):={[u,v]}_s\bigcup {[v,-u]}_s$

which are semicircles of various types passing through $v$.

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 $C\subset {\mathbf{S}}^{n-\mathrm{1}}$ is an s-convex set and $f:C\to \mathbf{R}$ is a function, then $f$ is called spherically convex if the function $f\circ \gamma :[a,b]\to \mathbf{R}$ is a univariate convex function for each minimal geodesic segment (function) $\gamma :[a,b]\to C$ (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 $C\subset {\mathbf{S}}^{n-\mathrm{1}}$ be an s-convex set. A function $f:C\to \mathbf{R}$ is called spherically convex (s-convex for brevity) if

$f(\lambda u_{\mathrm{1}}{+}_s(\mathrm{1}-\lambda )u_{\mathrm{2}})\le \lambda f(u_{\mathrm{1}})+(\mathrm{1}-\lambda )f(u_{\mathrm{2}})$

holds for $u_{\mathrm{1}},u_{\mathrm{2}}\in C$, $\lambda \in [\mathrm{0,1}]$ with $\lambda u_{\mathrm{1}}{+}_s\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}}\in C$.

For an s-convex function on $C$, it was shown in Ref.[19] that it is continuous with respect to the both metrics mentioned below, and also that restricted on each $k$-sphere contained in $C$$\left(\mathrm{0}\le k\le n-\mathrm{1}\right)$, it is a constant, in particular, $f(u)=f(-u)$ if $u,-u\in C$.

The intrinsic metric $d_s(\cdot ,\cdot )$ in ${\mathbf{S}}^{n-\mathrm{1}}$, defined by $d_s\left(u_{\mathrm{1}},u_{\mathrm{2}}\right):=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}〈\cdot ,\cdot 〉$ for $u_{\mathrm{1}},u_{\mathrm{2}}\in {\mathbf{S}}^{n-\mathrm{1}}$, will be used in studying the continuity and the differentiability of functions on ${\mathbf{S}}^{n-\mathrm{1}}$. An elementary geometric argument shows$\Vert u_{\mathrm{1}}-u_{\mathrm{2}} \Vert =\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{d_s(u_{\mathrm{1}},u_{\mathrm{2}})}{\mathrm{2}}$, so the intrinsic metric and Euclidean metric are equivalent on ${\mathbf{S}}^{n-\mathrm{1}}$.

For given $u\in {\mathbf{S}}^{n-\mathrm{1}}$ and $\delta >\mathrm{0}$, the set

$B_s(u,\delta ):=\{w\in {\mathbf{S}}^{n-\mathrm{1}}|d_s(u,w)<\delta \}$

is called an s-ball. A set $S\subset {\mathbf{S}}^{n-\mathrm{1}}$ is called s-open if for each $u\in S$, there is $\delta >\mathrm{0}$ such that $B_s(u,\delta )\subset S$. It is easy to see that $S$ is s-open if and only if there is an open set $\mathrm{\Omega }\subset {\mathbf{R}}^n$ such that $S=\mathrm{\Omega }\bigcap {\mathbf{S}}^{n-\mathrm{1}}$.

For $u_{\mathrm{1}},u_{\mathrm{2}}\in {\mathbf{S}}^{n-\mathrm{1}}$ with $u_{\mathrm{1}}\ne u_{\mathrm{2}}$, to describe the points in $[u_{\mathrm{1}},u_{\mathrm{2}}{]}_s$, there are two popular geodesic segment functions connecting $u_{\mathrm{1}},u_{\mathrm{2}}$ considered in Refs.[16, 17, 19] and Ref.[15], respectively:

$u_{\lambda }={\gamma }_{u_{\mathrm{1}},u_{\mathrm{2}}}(\lambda ):=\lambda u_{\mathrm{1}}{+}_s\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}},\lambda \in [\mathrm{0,1}]\mathrm{a}\mathrm{n}\mathrm{d}$

$u_{\theta }^{\mathrm{*}}={\gamma }_{u_{\mathrm{1}},u_{\mathrm{2}}}^{\mathrm{*}}(\theta ):=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{1}}{+}_s\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{2}},\theta \in [\mathrm{0},\alpha ]$

where $\alpha =d_s(u_{\mathrm{1}},u_{\mathrm{2}})$. It can be easily checked that $\theta =d_s(u_{\theta }^{\mathrm{*}},u_{\mathrm{1}})$ (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 $u_{\mathrm{1}},u_{\mathrm{2}}\in {\mathbf{S}}^{n-\mathrm{1}}$ with $u_{\mathrm{2}}\ne \pm u_{\mathrm{1}}$. Then $u_{\theta }^{\mathrm{*}}=u_{\lambda }$ if and only if

$\lambda =\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}$

(or equivalently, $\mathrm{1}-\lambda =\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}$).

Consequently,

$\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{u}_{\mathrm{1}}{+}_s\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{u}_{\mathrm{2}}$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{1}}+\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{2}}$

for all $\theta \in [\mathrm{0},\alpha ]$.

Proof   Clearly, $u_{\theta }^{\mathrm{*}}=u_{\lambda }$ iff $d_s(u_{\lambda },u_{\mathrm{1}})=d_s(u_{\theta }^{\mathrm{*}},u_{\mathrm{1}})(=\theta )$. For brevity, denote ${\overline{u}}_{\lambda }=\lambda u_{\mathrm{1}}+\left(\mathrm{1}-\lambda \right)u_{\mathrm{2}}$. By an elementary geometric argument, we see that $d_s(u_{\lambda },u_{\mathrm{1}})=\theta$ if and only if (noticing$\Vert u_{\mathrm{1}}-u_{\mathrm{2}} \Vert =\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}$),

$\Vert u_{\mathrm{2}}-\overline{u}{ }_{\lambda } \Vert =\{\begin{array}{l}\Vert u_{\mathrm{2}}-\overline{u}{ }_{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.} \Vert +\Vert \overline{u}{ }_{\lambda }-\overline{u}{ }_{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.} \Vert \\ =sin\frac{\mathrm{\alpha }}{\mathrm{2}}+\mathrm{c}\mathrm{o}\mathrm{s}\frac{\mathrm{\alpha }}{\mathrm{2}}\mathrm{t}\mathrm{a}\mathrm{n}(\frac{\mathrm{\alpha }}{\mathrm{2}}-\mathrm{\theta }),\mathrm{\theta }\in [\mathrm{0},\frac{\mathrm{\alpha }}{\mathrm{2}}]\\ \Vert {\mathrm{u}}_{\mathrm{2}}-\overline{\mathrm{u}}{ }_{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.} \Vert -\Vert \overline{\mathrm{u}}{ }_{\mathrm{\lambda }}-\overline{\mathrm{u}}{ }_{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{2}$}\right.} \Vert \\ =sin\frac{\mathrm{\alpha }}{\mathrm{2}}-\mathrm{c}\mathrm{o}\mathrm{s}\frac{\mathrm{\alpha }}{\mathrm{2}}\mathrm{t}\mathrm{a}\mathrm{n}(\mathrm{\theta }-\frac{\mathrm{\alpha }}{\mathrm{2}}),\mathrm{\theta }\in [\frac{\mathrm{\alpha }}{\mathrm{2}},\mathrm{\alpha }]\end{array}=\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}-\mathrm{c}\mathrm{o}\mathrm{s}\frac{\alpha }{\mathrm{2}}\mathrm{t}\mathrm{a}\mathrm{n}(\theta -\frac{\alpha }{\mathrm{2}})$

and in turn if and only if

$\lambda =\frac{\Vert u_{\mathrm{2}}-\overline{u}{ }_{\lambda } \Vert }{\Vert u-v \Vert }=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}-\mathrm{c}\mathrm{o}\mathrm{s}\frac{\alpha }{\mathrm{2}}\mathrm{t}\mathrm{a}\mathrm{n}(\theta -\frac{\alpha }{\mathrm{2}})}{\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}}$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\frac{\alpha }{\mathrm{2}})-\mathrm{c}\mathrm{o}\mathrm{s}\frac{\alpha }{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}(\theta -\frac{\alpha }{\mathrm{2}})}{\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\frac{\alpha }{\mathrm{2}})}$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}(\mathrm{c}\mathrm{o}\mathrm{s}\frac{\alpha }{\mathrm{2}}\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}\theta )}$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{2}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\frac{\alpha }{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}\theta }$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{s}\mathrm{i}\mathrm{n}\theta -\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{s}\mathrm{i}\mathrm{n}\theta }$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}$

Finally, another elementary geometric argument (drawing a picture to check) shows that

$\Vert \overline{u}{ }_{\lambda (\theta )} \Vert =\frac{\mathrm{c}\mathrm{o}\mathrm{s}\frac{\alpha }{\mathrm{2}}}{\mathrm{c}\mathrm{o}\mathrm{s}(\frac{\alpha }{\mathrm{2}}-\theta )}=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{\mathrm{2}\mathrm{s}\mathrm{i}\mathrm{n}\frac{\alpha }{\mathrm{2}}\mathrm{c}\mathrm{o}\mathrm{s}(\frac{\alpha }{\mathrm{2}}-\theta )}$

$=\frac{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}$

Therefore,

$\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{u}_{\mathrm{1}}{+}_s\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\theta +\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{u}_{\mathrm{2}}$

$=u_{\lambda }=\frac{{\overline{u}}_{\lambda }}{\Vert {\overline{u}}_{\lambda }\Vert }=\frac{\mathrm{s}\mathrm{i}\mathrm{n}(\alpha -\theta )}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{1}}+\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{\mathrm{s}\mathrm{i}\mathrm{n}\alpha }{u}_{\mathrm{2}}$

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 ${\mathbf{S}}^{n-\mathrm{1}}$ .

First, we give the concepts of s-directional derivative and s-Gateaux differentiability.

Definition 3   Let $B_s(u_{\mathrm{0}},\delta )\subset {\mathbf{S}}^{n-\mathrm{1}}$ and $f:B_s(u_{\mathrm{0}},\delta )$$\to \mathbf{R}$ be a function. If for some $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$, the limit

$\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u_{\lambda })-f(u_{\mathrm{0}})}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}=:D_{s}f(u_{\mathrm{0}},v)$

exists, where $u_{\lambda }:=\lambda v{+}_s(\mathrm{1}-\lambda )u_{\mathrm{0}}$, then $D_{s}f(u_{\mathrm{0}},v)$ is called the s-directional derivative of $f$ at $u_{\mathrm{0}}$ along (direction) $v$.

If both $D_{s}f(u_{\mathrm{0}},v)$ and $D_{s}f(u_{\mathrm{0}},-v)$ exist and $D_{s}f(u_{\mathrm{0}},-v)=-D_{s}f(u_{\mathrm{0}},v)$, then $D_{s}f(u_{\mathrm{0}},v)$ is called the partial derivative of $f$ at $u_{\mathrm{0}}$ along (direction) $v$, denoted by $\frac{\partial f(u_{\mathrm{0}})}{{\partial }_{s}v}$.

If $D_{s}f(u_{\mathrm{0}},v)$ exists for all $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$, then $f$ is called s-Gateaux differentiable (s-G-differentiable for brevity) at $u_{\mathrm{0}}$.

Remark 2   i) $\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})$ in Definition 1 can be replaced by $d_s(u_{\lambda },u_{\mathrm{0}})$ since $\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}{d_s(u_{\lambda },u_{\mathrm{0}})}=\mathrm{1}$.

ii) Naturally, for each nonzero $w\in <u_{\mathrm{0}}{>}^{\perp }$, one may define as well $D_{s}f(u_{\mathrm{0}},w)=\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u_{\lambda })-f(u_{\mathrm{0}})}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}$(if exists), where $u_{\lambda }:=\lambda w{+}_s(\mathrm{1}-\lambda )u_{\mathrm{0}}$. However, we point out that such an extended definition is not essentially necessary: writing $w=tv$ for some $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$ and $t>\mathrm{0}$, we have by the property of $\rho$,

$\lambda w{+}_s(\mathrm{1}-\lambda )u_{\mathrm{0}}=\rho (\lambda tv+(\mathrm{1}-\lambda )u_{\mathrm{0}})$

$=\rho (\frac{\lambda t}{\lambda (t-\mathrm{1})+\mathrm{1}}v+\frac{\mathrm{1}-\lambda }{\lambda (t-\mathrm{1})+\mathrm{1}}{u}_{\mathrm{0}})=\lambda w{+}_s(\mathrm{1}-\lambda )u_{\mathrm{0}}$

where $\mu =\frac{\lambda t}{\lambda (t-\mathrm{1})+\mathrm{1}}$, and in turn $\lambda \to {\mathrm{0}}^{+}$ iff $\mu \to {\mathrm{0}}^{+}$.

Thus,$D_{s}f(u_{\mathrm{0}},w)$ exists iff $D_{s}f(u_{\mathrm{0}},v)$ exists, and in such a case $D_{s}f(u_{\mathrm{0}},w)=D_{s}f(u_{\mathrm{0}},v)$.

Consider the function $f:{\mathbf{R}}^n\to \mathbf{R}$ defined by $f(x):=d_{s}f(u_{\mathrm{0}},x)$$(x\in {\mathbf{S}}^{n-\mathrm{1}})$ or 0 (otherwise), where $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ is fixed. It is easy to check that, for each $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$, the (usual) directional derivative

$Df(u_{\mathrm{0}},v):=\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u_{\mathrm{0}}+\lambda v)}{\lambda }=\mathrm{0}$

while the s-directional derivative $D_{s}f(u_{\mathrm{0}},v)=\mathrm{1}$. Also, in a same manner, one may construct a function $f$ such that $D_{s}f(u_{\mathrm{0}},v)$ (resp.$Df(u_{\mathrm{0}},v)$) exists, but $Df(u_{\mathrm{0}},v)$ (resp.$D_{s}f(u_{\mathrm{0}},v)$) does not. Therefore, the s-directional derivative and the (usual) directional derivative (along direction $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$) have nothing to do with each other in general. However, we have the following conclusion.

Proposition 1   Let $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ and $\mathrm{\Omega }$ be an open set containing $u_{\mathrm{0}}$. If a function $f:\mathrm{\Omega }\to \mathbf{R}$ is differentiable at $u_{\mathrm{0}}$, then for each $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$, $D_{s}f(u_{\mathrm{0}},v)$ exists and

$D_{s}f(u_{\mathrm{0}},v)=Df(u_{\mathrm{0}},v)$

Proof   We have $f(x)-f(u_{\mathrm{0}})=<\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}),x-u_{\mathrm{0}}>+$$o(\Vert x-u_{\mathrm{0}}\Vert )$ for $x\in \mathrm{\Omega }$ since $f$ is differentiable at $u_{\mathrm{0}}$, where $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f$ denotes the usual gradient of $f$.

Thus, to calculate $D_{s}f(u_{\mathrm{0}},v)$ for $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$, we compute $\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{u_{\lambda }-u_{\mathrm{0}}}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}$ first. Denoting $\theta :=d_s(u_{\lambda },u_{\mathrm{0}})$ (observing that $\theta \to {\mathrm{0}}^{+}$ iff $\lambda \to {\mathrm{0}}^{+}$), we have

$u_{\lambda }=\mathrm{s}\mathrm{i}\mathrm{n}\theta \cdot v+\mathrm{c}\mathrm{o}\mathrm{s}\theta \cdot u_{\mathrm{0}}$

and in turn

$\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{u_{\lambda }-u_{\mathrm{0}}}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}=\underset{\theta \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta \cdot v+(\mathrm{c}\mathrm{o}\mathrm{s}\theta -\mathrm{1})\cdot u_{\mathrm{0}}}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }=v$

Therefore, we obtain

$\begin{array}{l}{D}_{s}f(u_{\mathrm{0}},v)=\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u_{\lambda })-f(u_{\mathrm{0}})}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}\\ =\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{<\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}),u_{\lambda }-u_{\mathrm{0}}>+o(\Vert u_{\lambda }-u_{\mathrm{0}}\Vert )}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}\\ =<\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}),\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{u_{\lambda }-u_{\mathrm{0}}}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}>+\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{o(\Vert u_{\lambda }-u_{\mathrm{0}}\Vert )}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}\\ =<\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}),v>=D_{s}f(u_{\mathrm{0}},v)\end{array}$

where we used the fact $o(\Vert u_{\lambda }-u_{\mathrm{0}}\Vert )=o(\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}}))$ since $\underset{\lambda \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}{\Vert u_{\lambda }-u_{\mathrm{0}}\Vert }=\underset{\lambda \to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}{\mathrm{2}\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\frac{d_s(u_{\lambda },u_{\mathrm{0}})}{\mathrm{2}}}=\mathrm{1}$. The proof completes.

Next, we define the s-gradient of functions defined on ${\mathbf{S}}^{n-\mathrm{1}}$.

Definition 4   Let $B_s(u_{\mathrm{0}},\delta )\subset {\mathbf{S}}^{n-\mathrm{1}}$ and $f:B_s(u_{\mathrm{0}},\delta )$$\to \mathbf{R}$ be a function s-G-differentiable at $u_{\mathrm{0}}$. If $v^{\mathrm{*}}\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$ satisfies $D_{s}f(u_{\mathrm{0}},v^{\mathrm{*}})=\mathrm{m}\mathrm{a}\mathrm{x}\{D_{s}f(u_{\mathrm{0}},v)|v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }\}$ ,then $\left|D_{s}f(u_{\mathrm{0}},v^{\mathrm{*}})\right|v^{\mathrm{*}}$ is called an s-subgradient of $f$ at $u_{\mathrm{0}}$. If the s-subgradient is unique, then it is called the s-gradient of $f$ at $u_{\mathrm{0}}$, denoted by $\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})$.

Examples are easily found to show that, for a function defined in an open neighborhood of $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$, 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 $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ and $\mathrm{\Omega }$ be an open neighborhood of $u_{\mathrm{0}}$. If a function $f:\mathrm{\Omega }\to \mathbf{R}$ is differentiable at $u_{\mathrm{0}}$, then $\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})$ exists and

$\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})=P_{u_{\mathrm{0}}^{\perp }}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}))$

where $P_{u_{\mathrm{0}}^{\perp }}$ denotes the orthogonal projection from ${\mathbf{R}}^n$ to $<u_{\mathrm{0}}{>}^{\perp }$. In turn, $D_{s}f(u_{\mathrm{0}},v)=<\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}}),v>$ for $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$.

Proof   Writing

$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}})=P_{u_{\mathrm{0}}}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}))+P_{u_{\mathrm{0}}^{\perp }}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}))$

we have, as shown in the proof of Proposition 1, for each $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$,

$\begin{array}{l}{D}_{s}f(u_{\mathrm{0}},v)=<\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}),v>\\ =<P_{u_{\mathrm{0}}}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}))+P_{u_{\mathrm{0}}^{\perp }}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}})),v>\\ =<P_{u_{\mathrm{0}}^{\perp }}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}})),v>\end{array}$

since $v\in <u_{\mathrm{0}}{>}^{\perp }$, which implies clearly $\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})$ exists and $\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})=P_{u_{\mathrm{0}}^{\perp }}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f(u_{\mathrm{0}}))$, and in turn

$D_{s}f(u_{\mathrm{0}},v)=<\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}}),v>$

for all $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$.

Ferreira et al in Ref.[15] took $P_{u_{\mathrm{0}}}(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}f)$ 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 $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ and $f:U\to \mathbf{R}$ be a function, where $U$ is an s-open set containing $u_{\mathrm{0}}$. If there is $v°\in <u_{\mathrm{0}}{>}^{\perp }$, such that

$f(u)-f(u_{\mathrm{0}})=<v°,\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u,u_{\mathrm{0}})>+o(\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u,u_{\mathrm{0}}))$

whenever $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$ and $u\in [u_{\mathrm{0}},-u_{\mathrm{0}}{)}_s(v)\bigcap U$, then $f$ is called s-Frechet differentiable (s-F-differentiable for brevity or s-differentiable simply) at $u_{\mathrm{0}}$, and $f_s^{\mathrm{\text{'}}}(u_{\mathrm{0}}):=v°$ is called the s-F-derivative of$f$ at$u_{\mathrm{0}}$.

Remark 3   It is easy to check that if $f$ is differentiable at $u_{\mathrm{0}}$, then $f$ is s-differentiable at $u_{\mathrm{0}}$. However, the inverse is not true even if the function is defined in an open neighborhood of $u_{\mathrm{0}}$, as shown by the following example: consider the function $f:{\mathbf{R}}^n\to \mathbf{R}$ defined by

$f(x):=d_s^{\mathrm{2}}(u_{\mathrm{0}},x)$ ($x\in {\mathbf{S}}^{n-\mathrm{1}}$) or 1 (otherwise),

where $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ is fixed. It is easy to see that $f$ is s-differentiable but not differentiable at $u_{\mathrm{0}}$. 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 $f$ is defined on an s-open set $U$ containing $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ and s-differentiable at $u_{\mathrm{0}}$, then $\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}})=f_s^{\mathrm{\text{'}}}(u_{\mathrm{0}})$ and

$D_{s}f(u_{\mathrm{0}},v)=<\mathrm{g}\mathrm{r}\mathrm{a}{\mathrm{d}}_{s}f(u_{\mathrm{0}}),v>\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }.$

Proof   Since

$f(u_{\lambda })-f(u_{\mathrm{0}})=<f_s^{\mathrm{\text{'}}}(u_{\mathrm{0}}),\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})\cdot v>+o(\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}}))$

whenever $v\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$ and $u_{\lambda }:=\lambda v{+}_s(\mathrm{1}-\lambda )u_{\mathrm{0}}\in U$, we have

$D_{s}f(u_{\mathrm{0}},v)=\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u_{\lambda })-f(u_{\mathrm{0}})}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}$

$=<f_s^{\mathrm{\text{'}}}(u_{\mathrm{0}}),v>+\underset{\lambda \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{o(\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}}))}{\mathrm{s}\mathrm{i}\mathrm{n}{d}_s(u_{\lambda },u_{\mathrm{0}})}$

$=<f_s^{\mathrm{\text{'}}}(u_{\mathrm{0}}),v>$

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 $f$ is defined on an s-open $U$ containing $u_{\mathrm{0}}\in {\mathbf{S}}^{n-\mathrm{1}}$ and s-differentiable at $u_{\mathrm{0}}$, then for arbitrary pairly orthogonal $e_{\mathrm{1}},e_{\mathrm{2}},\cdots ,e_{n-\mathrm{1}}\in {\mathbf{S}}_{u_{\mathrm{0}}}^{\perp }$(that is, $e_{\mathrm{1}},e_{\mathrm{2}},\cdots ,e_{n-\mathrm{1}}$ form a standard orthogonal basis of $<u_{\mathrm{0}}{>}^{\perp }$), we have