Smulyan Lemma and Differentiability of the Support Function

Ildar Sadeqiand Sima Hassankhali

1 Department of Mathematics,Sahand University of Technology,Tabriz-Iran

2 Department of Mathematics,Sahand University of Technology,Tabriz-Iran

Abstract.The purpose of this paper is to verify the Smulyan lemma for the support function,and also the Gateaux differentiability of the support function is studied on its domain.Moreover,we provide a characterization of Frechet differentiability of the support function on the extremal points.

Key Words:Frechet and Gateaux differentiability,support function,strict convexity,Smulyan lemma.

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

where p is a positive integer,Rpis 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 σAcan be translated to a corresponding property of the cost function.See[11,12]and the long list of references therein.

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 σAis Gateaux differentiable on the interior of its domain int(domσA),which is an extension of[11,Theorem 6]into the infinite dimensional case.

2 Preliminaries

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,

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

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

and the domain of ?f is defined by

(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

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 hBand σAare 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 interiorandintA. By the separation theorem,there exists a nonzero bounded linear functional x?∈X?so thatBut the interior of A is dense in A(see[1,Lemma 5.28]),soThat is x?supports A at x. Lettingwe getandAlso,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:

(d1)A is strictly convex.

(d2)

(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

Also,

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],

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

Thus,

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

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

(e1)σAis Frechet differentiable atif and only ifwhenever xn,yn∈A satisfy limif and only ifis convergent whenever lim

(e2)σAis Gateaux differentiable atif and only ifwhenever xn,yn∈A satisfyif and only if there exists a unique x∈A which satisfies

(e3)hAis Frechet differentiable at x ∈X if and only ifwheneversatisfy limif and onlyis convergent whenever lim

(e4)hA0is Gateaux differentiable at x∈X if and only ifwhenever fn,gn∈A0satisfy lim fn(x)=limgn(x)=hA0(x)if and only if there exists a unique f ∈A0which satisfies

Remark 3.1.(f1)Differentiability conditions for σAand hA0are homogeneous,indeed hA0is differentiable at x if it is differentiable at λx for some scalar λ. Also σAis differentiable atif 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 ifIt means that?σA(x?)=A for everyand σAis not Gateaux differentiable at anyunless A is a singleton.Thus,when we speak about differentiability of σAon X?,we mean that it is differentiable on the set dom

(f4) Based on the fact thatfor every x ∈X and A ?X,from(e1)of Theorem 3.1,σAis Frechet differentiable atif and only if σA?xis Frechet differentiable atFrom(e2)of Theorem 3.1,σAis Gateaux differentiable atif and only if σA?xis Gateaux differentiable atAlso,for all y∈X.

Theorem 3.2.Let A be a closed convex neighborhood of zero.If A0is strictly convex,then hA0is Gateaux differentiable on int(domhA0){0}.

Proof.The support function hA0is 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),

Corollary 3.1.Let A be a closed strictly convex subset of the Banach space X with nonempty interior(int).Then,σAis Gateaux differentiable on the int(domσA){0}.Proof.Let z ∈intA.By(f4)of Remark 3.1,σAis Gateaux differentiable atif and only if σA?zis Gateaux differentiable at.So,without loss of generality,assume that 0 ∈A. From the Bipolar theorem,we get(A)00=A. Since A0is a neighborhood of zero,applying Theorem 3.2 for A0,we conclude that σAis 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.

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

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

and f0∈dom?σAwhich implies that σAis Gateaux differentiable at f0.So,from(e2)of Smulyan lemma,we get x=y.Therefore,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,σAis 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 σAis Gateaux differentiable on theif and only if A is r-strictly convex.

With applying Theorem 3.3 for C0the 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 σAis 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,σAis Gateaux differentiable onif and only if A is strictly convex.Also,when intwe have lin0A=X.Hence,and σAis Gateaux differentiable onif and only if A is rstrictly convex.

In[5],it is shown that for a bounded set A,a functionalstrongly exposes A if and only if the support function σAis Frechet differentiable atIf 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 pointstrongly exposes A?X if and only if the support function σAis Frechet differentiable at

Proof.Let x∈X.Based on(e4)of Remark 3.1,σAis Frechet differentiable atif and only if σA?xis Frechet differentiable atAlsostrongly exposes A,if and only ifstrongly exposes A?x.So,we may assume that 0∈A.

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

Therefore,it is enough to prove the theorem for Minkowski functional PA0. From[5,Corollary 7.20],strongly exposes F on A00if and only if σAis 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 σAis Gateaux differentiable on

Proposition 4.1.Let A be a nonempty closed,bounded and convex subset of a reflexive Banach space X andThen,σAis 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

then σAis 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.IfThen σAis Gateaux differentiable on int(domσA)if and only if(4.1)holds.

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 σAis Gateaux differentiable on int(domσA).