Pavel V. Semenov

Ernest Michael and theory of continuous selections

1. IntroductionFor a large number of those working in topology, functional analysis, multivalued analy-sis, approximation theory, convex geometry, mathematical economics, control theory, andseveral other areas, the year 1956 has always been strongly connected with the publicationby Ernest Michael of two fundamental papers on continuous selections which appeared inthe Annals of Mathematics [4] [5].With sufficient precision that year marked the beginning of the theory of continuousselections of multivalued mappings. In the last fifty years the approach to multivaluedmappings and their selections, set forth by Michael [4] [5], has well established itself incontemporary mathematics. Moreover, it has become an indispensable tool for manymathematicians working in vastly different areas.Clearly, the principal reason for this is the naturality of the concept of selection. Infact, many mathematical assertions can be reduced to using the linguistic reversal “∀x∈X ∃y∈ Y...”. However, as soon as we speak of the validity of assertions of the type∀x∈ X ∃y∈ Y P(x,y)it is natural to associate to every xa nonempty set of all those yfor which P(x,y) is true.In this way we obtain a multivalued map which can be interpreted as a mapping, whichassociates to every initial data x∈ Xof some problem P a nonempty set of solutions ofthis problemF: x→ {y∈ Y : P(x,y)}, F: X→ Y.The question of the existence of selections in such a setting turns out to be the questionabout the unique choice of the solution of the problem under given initial conditions.Different types of selections are considered in different mathematical categories.

Examples and Counterexamples

The aim of this chapter is to show that the principal properties of values of lower semicontinuous mappings (closedness, convexity,...) are essential in the main selection theorems of §1–§5. In Theorems (6.1), (6.4), (6.5) and (6.10) we follow (with modifications) [258]. Theorem (6.8) is taken from [271] (for another proof see [79]). Example from Theorem (6.7) was constructed in [262]. The remarkable example due to Pixley [331] is the last theorem (6.13) of this chapter and of the Theory.

HEREDITARY INVERTIBLE LINEAR SURJECTIONS AND SPLITTING PROBLEMS FOR SELECTIONS

Abstract Let A + B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h : X → A + B splits into a sum h = f + g of continuous mappings f : X → A and g : X → B ? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.

On continuous choice of retractions onto nonconvex subsets

Abstract For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A ∈ A can be chosen to depend continuously on A , whenever nonconvexity of each A ∈ A is less than 1 2 . The key geometric argument is that the set of all uniform retractions onto an α -paraconvex set (in the spirit of E. Michael) is α 1 − α -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate α 1 − α can be improved to α ( 1 + α 2 ) 1 − α 2 and the constant 1 2 can be replaced by the root of the equation α + α 2 + α 3 = 1 .

On controlled extensions of functions

For any numerical function E : R 2 → R we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions f : A → R, g : A → R and h : X → R satisfy the equality E(f (a), g(a)) = h(a) ,f or everya ∈ A ,t hen we are interested to find the extensions ˆ f and ˆ g of f and g, respectively, such that E( ˆ f( x),ˆ g(x)) = h(x), for every x ∈ X. We generalize earlier results concerning E(u,v) = u · v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.

On the Lebesgue function of open coverings

Abstract An approximation theorem for an upper semicontinuous mapping F from an arbitrary (not necessarily ANR) metric space X is proved, under certain control on degree of nonconvexity of values F ( x ), x ∈ X . The proof uses a generalization of the lemma on the Lebesgue number of a covering for the noncompact case.

On Strong Approximations of USC Nonconvex-Valued Mappings

For any upper semicontinuous and compact-valued (usco) mapping F : X?Y from a metric space X without isolated points into a normed space Y we prove the existence of a single-valued continuous mapping f : X?Y such that the Hausdorff distance between graphs ?F and ?f is arbitrarily small, whenever “measure of nonconvexity” of values of F admits an appropriate common upper estimate. Hence, we prove a version of the Beer?Cellina theorem, under controlled withdrawal of convexity of values of multifunctions. We also give conditions for such strong approximability of star-shaped-valued uppersemicontinuous (usc) multifunctions in comparison with Beers result for Hausdorff continuous star-shaped-valued multifunctions.

Selection Theorems for Non-Lower Semicontinuous Mappings

While lower semicontinuity of a mapping with closed convex values is sufficient for the existence of continuous selections, it is, of course, not necessary. For example, one can start by arbitrary continuous singlevalued map f : X→Y and then define F(x) to be a subset of Y such that f (x) ∈ F(x). Then f is a continuous selection for F, but there are no continuity type restrictions for F.

Convex-Valued Selection Theorem

This chapter deals more or less with a single theorem — the one stated in the title. This theorem gives sufficient conditions for the solvability of the continuous selection problem for a paracompact domain. But in order to introduce paracompactness from a selection point of view we start by searching for the necessary condition for the existence of such a solution. In Section 1 we prove that the existence of continuous selections of lower semicontinuous mappings with closed convex values implies the existence of locally finite refinements and locally finite partitions of unity. In Sections 2 and 3 we present two approaches to proving the Convex-valued selection theorem. In Section 2 the answer is given as a uniform limit of continuous e n -selections, whereas in Section 3, it is given as a uniform limit of e n -continuous selections. In (auxiliary) Section 4 we prove the equivalence of definitions of paracompact spaces via coverings and via partitions of unity. Also, we collect there the material concerning the properties of paracompact spaces, nerves of coverings and some facts about dimension theory.

Addendum: New Proof of Finite-Dimensional Selection Theorem

The main goal of this chapter is to present a new approach to the proof of the Finite-dimensional selection theorem (see §5) which was recently proposed by Scepin and Brodskiĭ [373]. First, note that there is only one proof of the Finite-dimensional selection theorem [259]. (Observe that the proof [131] is a reformulation of Michael’s proof in terms of coverings and provides a way to avoid uniform metric considerations.) Second, [373] gives in fact a generalization of Michael’s theorem. Third, and most important, this approach is based on the technique, which is widely exploited in other branches of the theory of multivalued mappings. Namely, in the fixed-point theory, where proofs are often based on UV n -mappings and on (graph) approximations of such mappings (see [16]).