#### Abstract

We first present a generalization of -Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space .

#### 1. Introduction

The infinite Rademacher theorem [1, 2] states that every Lipschitz mapping from a separable Banach space to a Banach space with the Radon-Nikodým property (in short, RNP) is almost everywhere Gâteaux differentiable, i.e., everywhere Gâteaux differentiable off an Aronszajn null set. It is shown in [3, 4] that Aronszajn null sets (introduced by Aronszajn [1]), Gauss null sets (introduced by Phelps [5]), and cube null sets (introduced by Mankiewicz [6]) coincide. In particular, Aronszajn null sets and Lebesgue null sets coincide in a finite dimensional Banach space [3]. To establish the localized version of the infinite Rademacher theorem, Cheng and Zhang [7] substituted closed convex subsets of a Banach space for and proved that every Lipschitz mapping from a closed convex subset of a Banach space with nonempty nonsupport point set to a Banach space with the RNP is almost everywhere Gâteaux differentiable.

As is well known, dual Banach spaces without separability may fail RNP. The Lipschitz mappings from a separable Banach space to a dual Banach space without separability need not be Gâteaux differentiable anywhere. To circumvent this obstruction, a weaker notion of a -Gâteaux differentiability was introduced (see, for instance, [3, 8]). Heinrich and Mankiewicz [9] proved the infinite -Rademacher theorem: every Lipschitz mapping from a separable Banach space to the dual space of a separable Banach space is almost everywhere -Gâteaux differentiable. Therefore, a question naturally arises.

*Question 1. *The question is whether the infinite -Rademacher theorem can be localized to those Lipschitz mappings from a closed convex set admitting nonempty nonsupport point set to the dual space of a separable Banach space.

The aim of this paper focuses on the study of the question above. Based on the ideas of the infinite -Rademacher theorem, we shall prove the following theorem.

Theorem 2. *Suppose that is a closed convex set of a separable Banach space with and that is a separable Banach space. Let be a Lipschitz mapping. Then is -Gâteaux differentiable off a Aronszajn null set of .*

As an application of Theorem 2, we obtain the following result.

Theorem 3. *Every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space.*

With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally get the following result.

Theorem 4. *Every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space .*

The letter will always be a real Banach space and its dual. denotes the unit sphere of . For a subset , we denote by the closure of . For simplicity, we also denote and for some .

#### 2. A Proof of Theorem 2

We first recall definitions of support points [5, 7] and Aronszajn null sets [1, 3].

*Definition 5. *Suppose that is a convex set in a Banach space . A point is said to be a support point of if there exists a non-zero functional such that .

We denote by the set of all support points of and by the set of all nonsupport points of .

Let be a Banach space. Let be the one dimensional Lebesgue measure. For each , setand for a finite or infinite sequence of nonzero points in , set

*Definition 6. *Suppose that is a Borel set in a separable Banach space . The set is called an Aronszajn null set (or simply, a null set) if

Note that is a set; hence it is a Borel set (see, for instance, [7, 10]).

The following two lemmas are presented in [3, 8].

Lemma 7. *Let be a Banach space with the RNP. Then every Lipschitz function from an open subset of into is Gâteaux differentiable off a null set of .*

Lemma 8. *Let be an -dimensional subspace of Banach space , and let be a basis for . Let be the Lebesgue measure on , and let be a Borel subset of such that for every . Then .*

*Definition 9. *Suppose that is a closed convex set of a separable Banach space with and that is a separable Banach space. Let be a Lipschitz mapping. Then is said to be -Gâteaux differentiable at if there exists a bounded linear operator such that for every and every the following limit exists:In this case is called the -derivative of at and is denote by .

Theorem 10. *Suppose that is a closed convex set of a separable Banach space with and that is a separable Banach space. Let be a Lipschitz mapping; i.e., there exists such that for all . Let be a sequence which consists of linearly independent vectors in . For every , set and set . Then**(i) is -Gâteaux differentiable off a null set of for every ;**(ii) whenever exists for every ;**(iii) If, in addition, there exists such that for all , then then is a linear isomorphism for almost every .*

*Proof. *(i). Without loss of generality, we assume that . Let be a dense sequence in . Given , define for every and . Then are Lipschitz real-valued functions on . Since is a sequence which consists of linearly independent vectors in and , we obtain that , and hence, is nonempty open in (see also p.p, 9 in [11]).

Applying Lemma 7 to and , then are Gâteaux differentiable off a null set of . Let denote the derivative of at .

Consider the set Then is not null set in . Choose ; then the following limits exists:for every , . Since is Lipschitz, is a bounded -Cauchy sequence when . By Alaoglu theorem, - exists in . Since is linear for every , the limit - is linear, and hence it is a bounded linear operator from into because is Lipschitz. Therefore, exists for every and .

(ii). For every , and , we haveThis completes our assertion (ii).

(iii). We omit the proof because the proof of assertion (iii) is similar to the proof of assertion (c) of Theorem 14.2.18, p. 385 in [8].

*Proof of Theorem 2. *Without loss of generality, we assume that . By Theorem 2.6 in [7], is a dense subspace of . Since is separable, there exists a sequence of linearly independent vectors in such that is dense in . For each , set . Letand let . Since is a Borel set, so is . Therefore, for each ,defines a bounded linear operator from to .

Since is dense in , there exists a unique bounded linear extension of from to . Therefore, and is just the set of all -Gâteaux differentiability points of in . It remains to show that D is not null.

By Theorem 10, is a null set in . Given , we define by . Then is a Lipschitz mapping on set , and by Lemma 7, is the set of all non-Gâteaux differentiability points of . By Lemma 8, . Therefore, .

Next, for any sequence of non-zero vectors in whose linear span is dense in , we shall prove . If , we just repeat the procedure above. Otherwise, for any such that , if for every , then there exists such that . Therefore, . This implies that is null set.

#### 3. Fixed Point Property for Isometries

In this section we apply -Gâteaux differentiability of Lipschitz mappings to the fixed point theory. As a result, we obtain Theorem 3.

Let be a nonempty bounded closed convex subset of a Banach space . Recall that a mapping is said to be isometry if whenever . We say that has the fixed point property for isometries if every isometry has a fixed point.

*Definition 11. *Suppose that is a subset of a Banach space and that is a Banach space. We say that Lipschitz embeds into provided that there is a mapping and constants such that for all , In this case, is said to be Lipschitz embedding and the smallest possible constant is called the Lipschitz constant of the mapping, in short, Lip

Proposition 12. *Suppose that is a closed convex separable set in a Banach space . Then in .*

*Proof. *We may assume that . Since is separable, is a separable space. It follows immediately from Proposition 1 in [10] that .

Theorem 13. *Suppose that is a separable closed convex set of a Banach space , and that is a separable Banach space. Then Lipschitz embeds into if and only if is linearly isomorphic to a subspace of .*

*Proof. **Sufficiency*. It is clearly trivial.*Necessity*. From Proposition 12, in . We may assume that . It follows from Theorem 2.6 in [7] that is a dense subspace of . Note that is separable, thus there exists a sequence of linearly independent vectors in such that is dense in . Let be a Lipschitz embedding. For each , set . Consider the set of all points for which there is a bounded linear operator such thatandFrom the proof of Theorem 10, the conclusion follows using argument similar to the proof of Theorem 2.

Proposition 14. *Suppose that is a separable closed convex set of a Banach space , and that is a dual Banach space. Then Lipschitz embeds into if and only if is linearly isomorphic to a subspace of .*

*Proof. *It suffices to show the necessity. Let and let be a Lipschitz embedding. Since is a separable, is a separable subspace of . By Corollary 14.2.22 in [8], we can find a separable Banach space and linearly isometric embeddings and . Hence, is a Lipschitz embedding. By Theorem 13, is linearly isomorphic to a subspace of . Note that is linearly isometric isomorphic to a subspace of . Therefore, is linearly isomorphic to a subspace of .

*Proof of Theorem 3. *Let be a nonempty bounded closed convex subset of a Banach space and let be Lipschitz embedded into a super reflexive space . We still assume . We can also assume that is separable because the fixed point property for isometries of a bounded closed convex set in a Banach space is separably determined. Superreflexivity of entails and hence is also a super reflexive space. By Proposition 14, is linearly isomorphic to a closed subspace of . This implies is a super reflexive space. Therefore, has the fixed point property for isometries by Maurey’s fixed point theorem [12].

*Remark 15. *The converse Theorem 3 does not hold. In fact, for a nonempty uniformly convexifiable set of Banach space , has the fixed point property for isometries (See [13, Theorem 4.2]). However, Beauzamy [14] proved that there exists a uniformly convexifiable set in a Banach space which can not linearly embedded into a super reflexive space.

#### 4. Fixed Point Property for Continuous Affine Mappings

In this section, with the application of Baudier-Lancien-Schlumprecht’s theorem, we obtain Theorem 4.

Let be a nonempty bounded closed convex subset of a Banach space . Recall that a mapping is said to be affine if whenever and . We say that has the fixed point property for continuous affine mappings if every continuous affine mapping has a fixed point.

Let and be two Banach spaces and be a subset of , and let be a mapping. Set andWe say that is uniformly continuous if and for all . is said to be uniformly embedded into provided that there is a mapping such that is injective and both and are uniformly continuous.

Proposition 16. *Suppose that is a nonempty bounded closed convex subset of a Banach space . If affinely uniformly embeds into a reflexive space , then has the fixed point property for continuous affine mappings.*

*Proof. *Suppose that affinely uniformly embeds into a reflexive space . Then there exist an affinely uniformly continuous mapping . We may assume that . Defining a mapping for all is a linearly uniformly continuous mapping from to . Thus, is a reflexive space. It follows from Schauder-Tychonoff theorem [15] that has the fixed point property for continuous affine mappings.

*Remark 17. *The affinity in Proposition 16 is necessary. Mazur [16] proved that the unit ball of is uniformly homomorphic onto the unit ball of . It is easy to see that has the fixed point property for continuous affine mappings by Schauder-Tychonoff theorem [15] and does not have the fixed point property for continuous affine mappings by Theorems 3.2 in [17]. On the other hand, the Mazur theorem [16] also implies that weak compactness is usually not preserved under uniform embeddings.

Recall the construction of the Tsirelson spaces and originally designed by Tsirelson [18]. Let and . We denote if and if . Here we set and . We say that a sequence is admissible if . Let be the canonical basis of . For every , we put and then define inductively for PutThen a norm on and is defined to be the completion of with respect to the norm . We denote the dual of by which is nowadays usually referred to as . For more detail, we refer the reader to [19].

For any infinite subset of , let . For each , let , where denotes the cardinality of the set . Elements of will always be listed in an increasing order; i.e., for every , we assume that . Recall that the Hamming metric is defined by where . Note that the metric , can be seen as the graph metric on the Hamming graph over a countable alphabet, denoted or simply , where two vertices are adjacent if they differ in exactly one coordinate. Let be a bijective. Then the map defined by is a Lipschitz mapping with Lip (see, for instant, [20]).

*Proof of Theorem 4. *By Schauder-Tychonoff theorem [15], it suffices to show that is weakly compact. Suppose, to the contrary, that is not weakly compact. Then, by James’ theorem [21], there exist and a sequences such that for all ,Choose . Thenwhich implies thatFor each , define a map by . Then is a Lipschitz mapping with Lip.

Let be a uniform embedding from into . Then Lip. By Theorem 4.4 in [20], for all , there exists such thatfor all . This impliesIn particular, for all , choose such that . We obtainThis is a contradiction when is sufficiently large and which completes our proof.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The first author was supported by Educational Commission of Hubei Province of China, Grant no. B2018046. The corresponding author was supported by NSFC, Grant no. 11501108, and by NSFF, Grant no. 2015J01579, 1991 Mathematics Subject Classification, 47H10, 46B03, and 46G05.