1 Introduction

The problem of differentiability and subdifferentiability of a convex function on a Banach space X are important in the theory of optimization(specially in economics)and geometry of Banach spaces.Recently,this issue has been discussed for specific convex functions known as support functions.In fact,they play a fundamental role in the development of optimization and variational analysis.

In economics,maximization of linear functionals on the subsets of Banach spaces has special importance in optimizing the price and profit.Shephard’s lemma is one of the most important results in economics.It is also associated with the differentiability of the cost function(see[10])defined by

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where p is a positive integer,Rp is the p-dimensional Euclidean space and A is a subset of (the positive cone of Rp).

Let X be a Banach space and A be a subset of X. Support function of the set A is defined by

Clearly,when dimX=p,the cost function g is strongly related to the support function σA.In fact,any property of the support function σA can be translated to a corresponding property of the cost function.See[11,12]and the long list of references therein.

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This article is organized as follows.In Section 2,we present some preliminaries.In Section 3,we state Smulyan lemma forthe support function and we establish some results regarding Smulyan lemma on the Gateaux and Frechet differentiability of support function.In Section 4,we show that the support function σA is Gateaux differentiable on the interior of its domain int(domσA),which is an extension of[11,Theorem 6]into the infinite dimensional case.

Lemma 7.19 of[5]states that when A is a closed convex neighborhood of zero,for ε ≥0 and x0 ∈X,

2 Preliminaries

A convex extended-valued function f is proper if and only if domand for each x∈X[1].The subdifferential of a proper function f at x∈dom f is

Let U be an open subset of the Banach space X and f:U→R be a real valued function.We say that f is Gateaux differentiable at x∈U,if for every h∈X,

exists in R and is a linear continuous function at The functionalis then called the Gateaux derivative or Gateaux differential of f at x.If,in addition,the above limit is uniform at h∈SX(where SX denotes the unit sphere in X),we say that f is Frechet differentiable at x.See[4]for more details.

We recall that the domain of a convex extended-valued function is the set

Throughout this paper,is a real Banach space whose dual X∗is endowed with the dual norm,denoted also by .We consider A ⊂X a nonempty set.As usual,we denote the interior of A,the cone generated by A,the affine hull of A,the linear space parallel to affA,the relative interior of A(that is the interior of A with respect to affine hull of A),the relative boundary of A,the closure of A,the convex hull of A and the polar set of A by intA,coneA,affA,lin0 A,rintA,rbdA,clA,convA and A0,respectively.

(e4)hA0 is Gateaux differentiable at x∈X if and only if whenever fn,gn∈A0 satisfy lim fn(x)=limgn(x)=hA0(x)if and only if there exists a unique f ∈A0 which satisfies

(see[3]).For nonempty subsets A⊂X and B⊂X∗,we define the support function of the set A by

and the support function of the set B by

It is well-known that(the same equality holds for hB).Therefore,we assume that A and B are nonempty closed and convex sets. Moreover,for a nonempty closed convex subset B of X∗,

- (P9+ Q9)ψ 9 Q9= - P9ψ9- Q9(ψ9∓1) = - cψψ9 2/ mb- cψ∓φ(ψ9∓1)2/ ma，

and when the space is reflexive,for a nonempty closed convex subset A of X,

also σA(0)=A and hB(0)=B(see[1,5]for more details).It is well-known that a lower semi-continuous proper convex function f on X is continuous at x ∈dom f if and only if x∈int(dom f)(see[2,Proposition 4.1.5]).Also,for every x∈int(dom f)and f is Gateaux differentiable at x ∈int(dom f)if and only if ∂f(x)is a singleton[5,theorem 7.17].Note that hB and σA are lower semi-continuous proper convex support functions.Hence,they are continuous and subdifferentiable on the interior of their domains.

3 Strict convexity of a set and Smulyan lemma

Let A be a nonempty closed convex subset of the Banach space X with nonempty interiorand intA. By the separation theorem,there exists a nonzero bounded linear functional x∗∈X∗so that But the interior of A is dense in A(see[1,Lemma 5.28]),so That is x∗supports A at x. Lettingwe get and Also, for every y∈A.Hence,

This can be summarized as follows.

Lemma 3.1.Let A be a closed convex subset of a real Banach space X with nonempty interior(int).Then hA0(x)=1 for every x∈bdA,where bdA denotes the boundary points of A.

Definition 3.1.A nonempty subset A of a Banach space X is said to be r-strictly convex(strictly convex)if every relative boundary point of A(boundary point of A)is an extreme point.

Proposition 3.1.Let A be a closed convex subset of the Banach space X.Then,the following are equivalent:

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(d1)A is strictly convex.

(d2)

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In addition,if intthen(d1)and(d2)are equivalent to the following:

(d3)∀x,y∈bdA;

(d4)∀x,y∈bdA;

When A is a bounded neighborhood of zero,the following replaces(d4).

Proof.It is easy to check thatForlet x,y∈bdA and

where From Lemma 3.1,Hence,Since A is strictly convex and x,y ∈bdA=extA,where extA denotes the extreme points of A),we get Therefore,is not a support point[1]and for allSo,for each x∗∈A0.Hence,which is a contradiction.

On the other hand,let x∈bdAextA.Then,there exist y,z∈bdA so thatFrom Theorem 3.1,and from(d3),the contradiction y=z is obtained.

By Lemma 3.1,It remains to show thatLet x,y∈X satisfy

Also,

Therefore,Since A is a bounded neighborhood of zero with intthere exist β,γ>0 and x1,y1 ∈bdA so that βx=x1 and γy=y1.According to Lemma 3.1,which means β=γ.Hence,by equality(3.1):

and from(d3),x1=y1.The proof is complete,because β=γ.

Let A be a closed convex neighborhood of zero. By the Bipolar theorem,Therefore,according to[5,Theorem 7.18],

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Thus,

These results lead us to write the Smulyan lemma[5,Lemma 7.20]for σA and hA0,as follows.

Theorem 3.1.Let A be a closed convex neighborhood of zero.Then

(e1)σA is Frechet differentiable at if and only if whenever xn,yn ∈A satisfy limif and only if is convergent whenever lim

(e2)σA is Gateaux differentiable at if and only if whenever xn,yn∈A satisfyif and only if there exists a unique x∈A which satisfies

(e3)hA is Frechet differentiable at x ∈X if and only if whenever satisfy limif and only is convergent whenever lim

and the domain of ∂f is defined by

Remark 3.1. (f1)Differentiability conditions for σA and hA0 are homogeneous,indeed hA0 is differentiable at x if it is differentiable at λx for some scalar λ. Also σA is differentiable at if it is differentiable at λx∗for some scalar λ.Consequently,it is enough to check the differentiability at points of a bounded neighborhood of zero.

(f2)Let A(B)be a closed convex subset of X(X∗).It is easy to check that ∂σA(0)=A(∂σB(0)=B). So,σA(hB)is Gateaux differentiable at 0 if and only if A(B)is a singleton.In this case,σA(hB)is Gateaux differentiable on X∗(X).

(f3) It is clear that x∗∈X∗is constant on A if and only if It means that∂σA(x∗)=A for every and σA is not Gateaux differentiable at anyunless A is a singleton.Thus,when we speak about differentiability of σA on X∗,we mean that it is differentiable on the set dom

(f4) Based on the fact that for every x ∈X and A ⊂X,from(e1)of Theorem 3.1,σA is Frechet differentiable at if and only if σA−x is Frechet differentiable at From(e2)of Theorem 3.1,σA is Gateaux differentiable at if and only if σA−x is Gateaux differentiable at Also,for all y∈X.

Theorem 3.2.Let A be a closed convex neighborhood of zero.If A0 is strictly convex,then hA0 is Gateaux differentiable on int(domhA0){0}.

Proof.The support function hA0 is a lower semi-continuous proper convex function that is subdifferentiable on the interior of its domain. Suppose that x0 ∈int(domhA0)and f,g∈∂hA0(x0).From the equality(3.2),

Hence,f,g ∈A0 andBut,from Bishop-Phelps theorem[1,Lemma 7.7],every support points of A0 is a boundary point of A0.Therefore,and under the assumption of strict convexity of A0,we get f=g.Hence,the set ∂hA0(x)is a singleton and by(e2)of the Smulyan lemma,hA0 is Gateaux differentiable.

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Corollary 3.1.Let A be a closed strictly convex subset of the Banach space X with nonempty interior(int).Then,σA is Gateaux differentiable on the int(domσA){0}.Proof.Let z ∈intA.By(f4)of Remark 3.1,σA is Gateaux differentiable at if and only if σA−z is Gateaux differentiable at .So,without loss of generality,assume that 0 ∈A. From the Bipolar theorem,we get(A)00=A. Since A0 is a neighborhood of zero,applying Theorem 3.2 for A0,we conclude that σA is Gateaux differentiable on the

In[7],Klee showed that every separable nonreflexive Banach spacecan be equivalently renormed so that the new norm is Gateaux differentiable but its dual norm is not strictly convex.So the inverse of Theorem 3.2 is not true in general.

and f0 ∈dom∂σA which implies that σA is Gateaux differentiable at f0.So,from(e2)of Smulyan lemma,we get x=y.Therefore,A is strictly convex.

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Proof.Without loss of generality,we assume that 0∈A(Remark 3.1,(f4)).For the if part,let x,y and Applying the Separation theorem forand intA,we have a nonzero linear bounded functional f0 ∈X∗so thatIt follows that

Theorem 3.3.Let A be a nonempty closed convex subset of a Banach space X with nonempty interior. Then,σA is Gateaux differentiable on the dom∂σA{0}if and only if A is strictly convex.

Finally,let f ∈dom∂σA{0}and x,y∈∂σA(f).Then,(from the Eq.(3.2)).From the Bishop phelps theorem,every support point of A is a boundary point of A.Hence, A and under the assumption of strict convexity of A,we have x=y.Therefore,∂σA(f)is a singleton and from Smulyan lemma,σA is Gateaux differentiable at f.

Note that if a Banach space X and its closed subspace Y are generated by weakly compact sets,then Y is complemented in X.In particular,reflexive Banach spaces have this property[8].Using the latter,we have the following result.

Theorem 3.4.Let A be a nonempty closed convex subset of a reflexive Banach space X with nonempty relative interior.Then σA is Gateaux differentiable on the if and only if A is r-strictly convex.

Proof.Let intThen and rintA=intA.So,Theorem 3.3 completes the proof in this case.Now,let intand set X0=lin0 A.If A is a singleton,then X0={0}and both assertions are true.Let dimX0 ≥1.By a translation,we assume that 0∈affA and From reflexivity of the space,there exists a subspace X1 of X so that X=X0×X1.So,A:=A0×{0},where A0 is a closed convex subset of X0 with nonempty interior.Hence,rbdA=bdA×{0}and A is r-strictly convex if and only if A0 is strictly convex.Now,we claim that σA is Gateaux differentiable on the domif and only if σA0 is Gateaux differentiable on dom∂σA0{0}.To prove this claim,using the same argument as in[11,Theorem 1],we get

for eachand

With applying Theorem 3.3 for C0 the proof is complete.

Remark 3.2. (g1)Let A be a nonempty closed convex subset of a finite dimensional Banach space X.Zalinescu showed that σA is differentiable onif and only if A is r-strictly convex(see [11,Theorem 2]).In fact,Theorem 3.4 is a generalization of Zalinescu’s theorem in infinite dimensional case.

(g2)In Theorem 3.4,when A is compact,σA is Gateaux differentiable on if and only if A is strictly convex.Also,when intwe have lin0 A=X.Hence,and σA is Gateaux differentiable on if and only if A is rstrictly convex.

In[5],it is shown that for a bounded set A,a functional strongly exposes A if and only if the support function σA is Frechet differentiable at If we replace bounded sets with closed convex sets,the theorem still remains true.

Theorem 3.5.Let A be a closed convex subset of the Banach space X.A point strongly exposes A⊂X if and only if the support function σA is Frechet differentiable at

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Proof.Let x∈X.Based on(e4)of Remark 3.1,σA is Frechet differentiable at if and only if σA−x is Frechet differentiable at Also strongly exposes A,if and only if strongly exposes A−x.So,we may assume that 0∈A.

By the Bipolar theorem[5,Theorem 3.38],A=Aoo and:

Therefore,it is enough to prove the theorem for Minkowski functional PA0. From[5,Corollary 7.20], strongly exposes F on A00 if and only if σA is Frechet differentiable atThe Bipolar theorem again,shows that A=A00,which completes the proof.

4 Differentiability of σA on int(domσA)

Let A be a nonempty closed convex subset of a Banach space X such that

The natural question is that if σA is Gateaux differentiable on

Proposition 4.1.Let A be a nonempty closed,bounded and convex subset of a reflexive Banach space X and Then,σA is Gateaux differentiable onif and only if A is r-strictly convex.

Proof.Since A is a closed bounded convex subset of X,it is w-compact and from the James theorem[1,Theorem 6.36],every continuous linear functional attains its supremum on A.Hence,

Therefore,by Theorem 3.4,the proof is completed.

Let

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In[11],it is shown that rbdA and if X is a finite dimensional Banach space and A is an unbounded subset of X for which

then σA is differentiable on int(domσA)if and only if

What follows is a generalization of the above theorem in infinite dimensional case.

Theorem 4.1.Let A be a nonempty subset of a reflexive Banach space X.If Then σA is Gateaux differentiable on int(domσA)if and only if(4.1)holds.

Proof.Let and λ ∈(0,1)be such that Then,there exists x∗∈int(domσA)such that Thus,andIt follows that Therefore,But this contradicts Gateaux differentiability of σA.Now,assume that σA is not differentiable on x∗∈int(domσA). By the assumptions, Also,as we mentioned in preliminary section,for every x∗∈int(domσC).Hence,there exist x1,x2 ∈∂σC(x∗)so that Now,from the convexity of ∂σC(x∗),we get the contradiction

Let X be a finite dimensional Banach space and A be a subset of X so that intThen,the following two assertions are equivalent[11],

and

Since this equivalence remains true in infinite dimensional reflexive Banach spaces,we obtain the following result.

Corollary 4.1.When the conditions(4.2)or(4.3)hold,then σA is Gateaux differentiable on int(domσA).

Proof.Since

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