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In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

The present note is a corrigendum to the paper “Best approximation in polyhedral Banach spaces”, J. Approx. Theory 163 (2011) 1748–1771.

We prove a new variational principle which in particular does not assume the complete- ness of the domain. As an application we give a new, more natural, proof of the fact that a real valued Lipschitz function on an Asplund space has points of Fr´ echet differentiability.

We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example: 1. AssumeX is separable and non-reflexive and its unit sphere is covered by a sequence of balls\(\left\{ {C_i } \right\}_{i = 1}^\infty \) of radiusa

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some infinite-dimensional X. Previously, even for collections consisting of two functions this has been known for finite-dimensional X only (although for one function the answer is known to be affirmative in full generality). Our aims are achieved by introducing a new class of null sets in Banach spaces (called F-null sets), whose definition involves both the notions of category and measure, and showing that the required differentiability holds almost everywhere with respect to it. We even obtain existence of Frechet derivatives of Lipschitz functions between certain infinite-dimensional Banach spaces; no such results have been known previously. Our main result states that a Lipschitz map between separable Banach spaces is Frechet differentiable F-almost everywhere provided that it is regularly Gateaux differentiable F-almost everywhere and the Gateaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodym property are Gateaux differentiable F-almost everywhere. Moreover, Gateaux differentiability implies regular Gateaux differentiability with exception of another kind of negligible sets, so-called a-porous sets. The answer to the question is therefore positive in every space in which every r-porous set is F-null. We show that this holds for C(K) with K countable compact, the Tsirelson space and for all subspaces of co, but that it fails for Hilbert spaces.

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ⊂ l2 which contains a translate of any compact set in the unit ball of l2 and a Lipschitz isomorphismF of l2 onto l2 so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA⊂X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on l2 (or more generally on an Asplund space) has points of Frechet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Frechet differentiability provided that the w * closure of the set of its points of Gâteaux differentiability is norm separable.

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L p , 1 < p < $ \infty $ , is already isomorphic to a linear quotient of L p .

The paper treats several topics in infinite-dimensional convexity. (1) Tiling and proximinal sets in Banach spaces. (2) Structure of infinite-dimensional polytopes. (3) Approximating convex sets by such sets which have a dense set of extreme points.

This paper is a contribution to the general problem of differentiability of Lipschitz functions between Banach spaces. We establish here a result concerning the existence of derivatives which are in some sense between the notions of Gâteaux and Frechet differentiability.

Following results of Bourgain and Gorelik we show that the spaces l p , 1

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

This chapter discusses Γ‎ₙ-nullness of sets porous “¹at infinity” and/or existence of many points of Fréchet differentiability of Lipschitz maps into n-dimensional spaces. The results reveal a σ‎-porous set whose complement is null on all n-dimensional surfaces and the multidimensional mean value estimates fail even for ε‎-Fréchet derivatives. Previous chapters have established conditions on a Banach space X under which porous sets in X are Γ‎ₙ-null and/or the the multidimensional mean value estimates for Fréchet derivatives of Lipschitz maps into n-dimensional spaces hold. This chapter investigates in what sense the assumptions of these main results are close to being optimal.

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

This chapter treats results on ε‎-Fréchet differentiability of Lipschitz functions in asymptotically smooth spaces. These results are highly exceptional in the sense that they prove almost Frechet differentiability in some situations when we know that the closed convex hull of all (even almost) Fréchet derivatives may be strictly smaller than the closed convex hull of the Gâteaux derivatives. The chapter first presents a simple proof of an almost differentiability result for Lipschitz functions in asymptotically uniformly smooth spaces before discussing the notion of asymptotic uniform smoothness. It then proves that in an asymptotically smooth Banach space X, any finite set of real-valued Lipschitz functions on X has, for every ε‎ > 0, a common point of ε‎-Fréchet differentiability.

This chapter introduces the notion of porosity “at infinity” (formally defined as porosity with respect to a family of subspaces) and discusses the main result, which shows that sets porous with respect to a family of subspaces are Γ‎ₙ-null provided X admits a continuous bump function whose modulus of smoothness (in the direction of this family) is controlled by tⁿ logⁿ⁻¹ (1/t). The first of these results characterizes Asplund spaces: it is shown that a separable space has separable dual if and only if all its porous sets are Γ‎₁-null. The chapter first describes porous and σ‎-porous sets as well as a criterion of Γ‎ₙ-nullness of porous sets. It then considers the link between directional porosity and Γ‎ₙ-nullness. Finally, it tackles the question in which spaces, and for what values of n, porous sets are Γ‎ₙ-null.

This chapter presents the main results on Gâteaux differentiability of Lipschitz functions by recalling the notions of the Radon-Nikodým property (RNP) and null sets. The discussion focuses not only on the mere existence of points of Fréchet differentiability, but also, and often more important, on the validity of the mean value estimates. After considering the RNP of a Banach space, the chapter examines Haar and Aronszajn-Gauss null sets. It then analyzes the existence result for Gâteaux derivatives as well as the meaning of multidimensional mean value estimates. It also explains how, for locally Lipschitz maps of separable Banach spaces to spaces with the RNP, the condition for the validity of the multidimensional mean value estimate may be simplified.

This chapter demonstrates that the results about smallness of porous sets, and so also of sets of irregularity points of a given Lipschitz function, can be used to show existence of points of (at least) ε‎-Fréchet differentiability of vector-valued functions. The approach involves combining this new idea with the basic notion that points of ε‎-Fréchet differentiability should appear in small slices of the set of Gâteaux derivatives. The chapter obtains very precise results on existence of points of ε‎-Fréchet differentiability for Lipschitz maps with finite dimensional range. The main result applies when every porous set is contained in the unions of a σ‎-directionally porous (and hence Haar null) set and a Γ‎ₙ-null Gsubscript Small Delta set.

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of n-dimensional surfaces. The idea is to deform a flat surface passing through a point x (along which we imagine that a certain function f is almost affine) to a surface passing through a point witnessing that f is not Fréchet differentiable at x. This is done in such a way that certain “energy” associated to surfaces increases less than the “energy” of the functionf along the surface. The chapter also discusses linear operators and tensor products, various notions and notation related to Fréchet differentiability, and deformation of surfaces controlled by ω‎ⁿ. Finally, it proves some integral estimates of derivatives of Lipschitz maps between Euclidean spaces (not necessarily of the same dimension).

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

This chapter describes the modulus of smoothness of a function in the direction of a family of subspaces and the much simpler notion of upper Fréchet differentiability. It also considers the notion of spaces admitting bump functions smooth in the direction of a family of subspaces with modulus controlled by ω‎(t). It shows that this notion is related to asymptotic uniform smoothness, and that very smooth bumps, and very asymptotically uniformly smooth norms, exist in all asymptotically c₀ spaces. This allows a new approach to results on Γ‎-almost everywhere Frechet differentiability of Lipschitz functions. The chapter concludes by explaining an immediate consequence for renorming of spaces containing an asymptotically c₀ family of subspaces.

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

This book deals with the existence of Fréchet derivatives of Lipschitz functions from X to Y, where X is an Asplund space and Y has the Radon-Nikodým property (RNP). It considers whether every countable collection of real-valued Lipschitz functions on an Asplund space has a common point of Fréchet differentiability. It also examines the conditions under which all Lipschitz mapping of X to finite dimensional spaces not only possess points of Fréchet differentiability, but possess so many of them that even the multidimensional mean value estimate holds. Other topics include the notion of the Radon-Nikodým property and main results on Gâteaux differentiability of Lipschitz functions and related notions of null sets; separable determination and variational principles; and differentiability of Lipschitz maps on Hilbert spaces.

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with the theorem stating that every Lipschitz map of a Hilbert space to a two-dimensional space has points of Fréchet differentiability. This is followed by a lemma, which is stated in an arbitrary Hilbert space but whose validity in the general case follows from its three-dimensional version. The chapter then explains the proof of the theorem and of the lemma stated above. In particular, it considers two cases, one corresponding to irregular behavior and the other to regular behavior.

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.